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Question: What did the court in In re Ferguson conclude about the transformation prong of the Bilski test?

Read the following text and answer briefly.

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PATENT, TRADEMARK
& COPYRIGHT !
Reproduced with permission from BNA’s Patent,Trademark 11/20/09, 11/20/2009  Copyright ஽ 2009 by The Bu-reau of National Affairs, Inc  (800-372-1033) http://www bna com As the patent community anticipates a decision by the U S  Supreme Court on subject matter patentability, recent rulings by the Federal Circuit and the Board of Patent Appeals and Interferences suggest strategies for preparing method patent applications that will sur- vive the Federal Circuit’s ‘‘machine-or-transformation’’ test 
The Changing Landscape of Method Claims in the Wake of In re Bilski:What We Can Learn from Recent Decisions of Federal Courts and the Board ofPatent Appeals rulings on software-based and other business methodpatent applications 
On review before the high court is the en banc ruling ‘‘Pure’’ business methods are out  Algorithms by the U S  Court of Appeals for the Federal Circuit1 are out  Machines and data transformations that, in order to be eligible for patent protection, an in- ventive method must either be tied to a machine or re- While the patent community waits for the Supreme cite a transformation of an article 2 This ‘‘machine-or- Court’s decision in Bilski v  Kappos, No  08-964 (U S 
transformation’’ test replaced the Freeman-Walter- argued Nov  9, 2009) (79 PTCJ 33, 11/13/09), patent ap- Abele3 test and the ‘‘useful, concrete and tangible plicants seeking to write patentable claims are stuckwith trying to conform to the lower courts’ most recent 1 In re Bilski, 545 F 3d 943, 88 USPQ2d 1385 (Fed  Cir 
2008) (en banc) (77 PTCJ 4, 11/7/08) 
2 ‘‘The machine-or-transformation test is a two-branched Adriana Suringa Luedke and Bridget M  Hay- inquiry; an applicant may show that a process claim satisfies den are lawyers at Dorsey & Whitney, Min- § 101 either by showing that his claim is tied to a particular neapolis  Luedke can be reached at machine, or by showing that his claim transforms an article ’’ leudke adriana@dorsey com  Hayden can be reached at hayden bridget@dorsey com  3 In re Freeman, 573 F 2d 1237, 197 USPQ 464 (C C P A 
1978); In re Walter, 618 F 2d 758, 205 USPQ 397 (C C P A 
COPYRIGHT ஽ 2009 BY THE BUREAU OF NATIONAL AFFAIRS, INC 
result’’ inquiry advocated in State Street,4 each of gregating, and selling real estate property and claims which had been applied by the Federal Circuit and its reciting a method of performing tax-deferred real estate predecessor court in various cases, and both of which property exchanges were not statutory under Section 101  Since no machine was recited, the only issue be- In this article, we examine the 2008 decision of the fore the court was whether the claims met the ‘‘trans- Federal Circuit, federal district court decisions, and de- formation’’ prong of the Bilski test 13 The court held cisions of Patent and Trademark Office’s Board of that the claims ‘‘involve[d] only the transformation or Patent Appeals and Interferences  Based upon the out- manipulation of legal obligations and relationships’’ comes in these cases, we offer guidance as to what is that did not qualify under Bilski 14 patent-eligible under 35 U S C  § 101, strategies for pre- Concerning the recitation of the ‘‘creation of deed- senting methods in patent applications and claiming shares’’ in some of the claims, the court found that the these methods, and possible ‘‘fixes’’ for applications deedshares themselves were not physical objects, but drafted pre-Bilski that must now withstand scrutiny un- only represented intangible legal ownership interests in der the new machine-or-transformation test 
property 15 Therefore, the creation of deedshares wasnot sufficient to establish patent eligibility under Bil- A number of recent federal court and board decisions have applied the patent eligibility test set forth in Bilski, implemented step to an otherwise obvious method was not sufficient to avoid invalidity of the claim  In KingPharmeuticals Inc  v  Eon Labs Inc ,17 the district court held invalid claims to a method of increasing the oral Several cases have addressed (and rejected) claims bioavailability of metaxalone because the claims were obvious over the prior art asserted by the accused in- In In re Ferguson,6 the Federal Circuit reviewed the board’s rejection of claims directed to a method of mar- Two dependent claims added a step of informing the keting a product and a ‘‘paradigm’’ for marketing soft- patient of certain results, which the patentee argued ware as nonstatutory subject matter under Section was not obvious  The court rejected this argument, con- 101 7 The appellate court affirmed the board’s rejection, cluding that ‘‘[b]ecause the food effect is an inherent concluding that the method claims were neither tied to property of the prior art and, therefore, unpatentable, a particular machine or apparatus nor did they trans- then informing a patient of that inherent property is form a particular article into a different state or thing 8 The court defined a machine broadly as ‘‘a concrete The court also commented that the added step of in- thing, consisting of parts, or of certain devices or com- forming the patient did not meet the patent eligibility binations of devices,’’ which did not include the ‘‘shared standard set forth in Bilski because the step did not re- marketing force’’ to which the method claims were quire use of a machine or transform the metaxalone into a different state or thing 19 Notably, this conclusion The claims directed to a ‘‘paradigm’’ were non- runs counter to the Supreme Court’s instruction that statutory because the claims did not fall within any of claims are to be examined ‘‘as a whole’’ and not dis- the four statutory categories (machines, manufactures, sected into old and new elements and that are evaluated compositions of matter and processes)  Concerning the two closest possible categories, the court concluded Recent board decisions have been consistent with the that the claimed paradigm was not a process, because holdings of the federal courts  For example, in Ex parte no act or series of acts was required, and was not a Roberts,21 the board found ineligible under Section 101 manufacture, because it was not a tangible article re- a ‘‘method of creating a real estate investment instru- sulting from a process of manufacture 10 Concerning ment adapted for performing tax-deferred exchanges’’ the recitation of a ‘‘marketing company’’ in the para- because the claim did not satisfy either the machine or digm claims, the court concluded that the patent appli- cants did ‘‘no more than provide an abstract idea—a Similarly, in Ex parte Haworth,23 a method for ‘‘at- business model for an intangible marketing com- tempting to collect payments from customers having delinquent accounts concurrently with a partner that In Fort Properties Inc  v  American Master Lease owns the delinquent accounts’’ was found to be patent LLC,12 the California district court held that claims re- ineligible because the claim wording was ‘‘broad in that citing a series of transactions involving acquiring, ag- 1980); In re Abele, 684 F 2d 902, 214 USPQ 682 (C C P A 
4 State Street Bank & Trust Co  v  Signature Financial 16 See Ex parte Roberts , 2009-004444 at 4-5 (B P A I  June Group, 149 F 3d 1368, 1370, 47 USPQ2d 1596 (Fed  Cir  1998) 19, 2009) (holding a ‘‘method of creating a real estate invest- ment instrument adapted for performing tax-deferred ex- changes’’ patent ineligible as not passing the machine-or- 7 The court accepted the board’s definition of ‘‘paradigm’’ 17 593 F  Supp 2d 501 (E D N Y  2009) 
to mean ‘‘a pattern, example or model ’’ Id  at 1362 
20 See Diamond v  Diehr, 450 U S  175, 188 (1981) 
21 No  2009-004444 (B P A I  June 19, 2009) 
12 2009 WL 249205, *5 (C D  Cal  Jan  22, 2009) 
23 No  2009-000350 (B P A I  July 30, 2009) 
it refers generally to extending an offer, receiving an machine  Accordingly, the process claims       are not acceptance, and paying a commission’’ and did not in- voke, recite or limit the method of implementation us-ing any particular machine or apparatus 24 The court also evaluated similar claims that recited the use of a ‘‘comparator’’ to perform the recited pixel- B  Software Claims Not Expressly Tied to a ‘Particular by-pixel comparison and held that this recitation also did not mandate a machine 29 While the court acknowl-edged that software was offered as one ‘‘option,’’ the Other cases have addressed software methods where court concluded that the claimed function of the com- the claim language was either not expressly tied to com- parator could also be performed in one’s mind or on pa- puter hardware components or the ties to computer per such that a machine was not required  The court components were somewhat ambiguous  In several further noted that, even though the ‘‘comparator’’ was cases, courts have rejected the recitation of generic defined as a ‘‘device,’’ ‘‘the use of the term ‘device’ is computer components as sufficient to satisfy the ‘‘ma- not synonymous with machine ’’30 As a result, none of chine’’ prong of the Bilski test  A number of these deci- the claims at issue met the ‘‘machine’’ prong of the Bil- sions also addressed the ‘‘transformation’’ prong of the Concerning the ‘‘transformation’’ prong, the court re- In Research Corporation Technology Inc  v  Mi- lied in particular upon the Abele decision in expanding crosoft Corp ,25 the district court considered the patent the requirements of this test by requiring that the eligibility of method claims in six patents directed to claimed transformation process be both ‘‘(1) limited to methods of halftoning of gray scale images by using a transformation of specific data, and 2) limited to a vi- pixel-by-pixel comparison of the image against a blue sual depiction representing specific objects or sub- noise mask  Relying on the Federal Circuit’s Bilski stances ’’31 It then concluded that a number of the analysis as well as a decision of its predecessor court, patent claims did not meet the second prong of this ex- In re Abele,26 the judge concluded that a number of the panded test because the claims did not ‘‘require any vi- sual depiction or subsequent display’’ even though the transformation test set forth in Bilski 27 claimed method did transform specific image data 32 Concerning the ‘‘machine’’ prong, the district court The district court also found other claims patent- found that the pixel-by-pixel comparison recited in the eligible under Section 101 because these claims recited claims did not require the use of a machine, but could the use of the comparison data ‘‘to produce a halftoned ‘‘dictate[d] a transformation of specific data, and [were] be done on a sheet of paper using a pen  The com- further limited to a visual depiction which represents parison uses formulas and numbers to generate a bi- specific objects ’’33 Thus, the patent eligibility of the nary value to determine the placement of a dot at a claims turned on whether the claims recited the use of location  Formulas and numbers not tied to a particu- the transformed data to generate a display 
lar machine cannot be patented, under the machine In DealerTrack Inc  v  Huber,34 the district court prong, even with a field-of-use limitation because granted a summary judgment of invalidity under § 101 they represent fundamental principles, and to do so of patent claims directed to ‘‘a computer aided method’’ would preempt the entire field  The patent claims      
of managing a credit application reciting the following do not mandate the use of a machine to achieve their algorithmic and algebraic ends  Simply because adigital apparatus such as a computer, calculator, or [A] receiving credit application data from a remote the like could assist with this comparison does not render it patent eligible material  RCT’s argument [B] selectively forwarding the credit application data that a pixel by its nature is electronic and therefore to remote funding source terminal devices; necessitates a machine is a post solution argumentand the Court rejects it  The claim construction specifies that the comparison is of a value to a mask 29 The term ‘‘comparator’’ was construed by the court to be (or set of values) to determine whether the dot is a ‘‘device (or collection of operations, as in software) that com- turned on at a specific location  This process does pares an input number (called the operand) to a number pre- not require a particular machine  The Bilski test is stored in the comparator (called the threshold) and produces clear: the process claims must be tied to a particular as output a binary value (such as ‘‘0,’’ zero) if the input is alge-braically less than the threshold [the result of comparing anoperand against a fixed threshold and setting an operand less 24 Id  at 9-10  See also, e g , Ex parte Farnes, 2009-002770 than the threshold to one value and an operand greater than (B P A I  June 2, 2009) (rejecting a method claim for develop- or equal to the threshold to another value], and produces the ing a solution to a customer experience issue including steps opposite binary value (such as ‘‘1,’’ one) if the input is algebra- of: ‘‘identifying a target customer,’’ ‘‘defining a current cus- ically greater than or equal to the threshold ’’ Id  at *17 (em- tomer experience,’’ ‘‘summarizing values and benefits’’ to pro- vide to the customer, and ‘‘identifying metrics for measuring success’’); Ex parte Salinkas, 2009-002768 (B P A I  May 18, 31 Id  at *9  Notably, Bilski concluded that the Abele visual 2009) (finding patent ineligible a method of launching a depiction was ‘‘sufficient’’ to establish transformation (545 knowledge network involving ‘‘selecting an executive spon- F 3d at 963), while the Research Corporation court went fur- sor,’’ ‘‘forming a core team of experts,’’ and ‘‘providing pre- ther by making visual depiction ‘‘required’’ to establish trans- 25 2009 WL 2413623 (D  Ariz  July 28, 2009) (78 PTCJ 432, 26 684 F 2d 902, 214 USPQ 682 (C C P A  1982) 
34 2009 WL 2020761 (C D  Cal  July 7, 2009) (78 PTCJ 341, PATENT, TRADEMARK & COPYRIGHT JOURNAL [C] forwarding funding decision data from at least tation of ‘over the Internet’ suffices to tie a process one of the remote funding source terminal de- claim to a particular machine’’ and concluded that it vices to the remote application entry and display The internet continues to exist despite the addition [D] wherein the selectively forwarding the credit ap- or subtraction of any particular piece of hardware  It may be supposed that the internet itself, rather than [E] sending at least a portion of a credit application any underlying computer or set of computers, is the to more than one of said remote funding sources ‘‘machine’’ to which plaintiff refers  Yet the internet is an abstraction  If every computer user in the world [F] sending at least a portion of a credit application unplugged from the internet, the internet would to more than one of said remote funding sources cease to exist, although every molecule of every ma- sequentially until a finding [sic ] source returns a chine remained in place  One can touch a computer or a network cable, but one cannot touch ‘‘the inter- [G] sending       a credit application       after a prede- Additionally, the court found that the recitation of the [H] sending the credit application from a first remote internet in this case merely constituted ‘‘insignificant funding source to a second remote funding extra-solution activity’’ and therefore did not qualify as a ‘‘particular machine’’ under Bilski 41 ‘‘[T]ossing in In concluding that the claim did not satisfy the Bilski references to internet commerce’’ was not sufficient to machine-or-transformation test, the court held that the render ‘‘a mental process for collecting data and weigh- claimed central processor, remote application and dis- ing values’’ patent-eligible 42 Additionally, ‘‘limiting’’ play device, and remote funding source terminal device the claim to use over the Internet was not a meaningful could be ‘‘any device’’ and did not constitute a ‘‘’par- limitation, such that the claims ‘‘broadly preempt the ticular machine’ within the meaning of Bilski ’’35 The fundamental mental process of fraud detection using court relied upon several board decisions to support its associations between credit cards ’’43 premise that ‘‘claims reciting the use of general purpose processors or computers do not satisfy the test ’’36 claim,44 notwithstanding the Federal Circuit’s holding In Cybersource Corp  v  Retail Decisions Inc ,37 the in In re Beauregard,45 the district court concluded that district court held claims for ‘‘a method for verifying the ‘‘there is at present no legal doctrine creating a special validity of a credit card transaction over the Internet’’ ‘‘Beauregard claim’’ that would exempt the claim from and ‘‘a computer readable medium containing program the analysis of Bilski ’’ Moreover, ‘‘[s]imply appending instructions for detecting fraud in a credit card transac- ‘A computer readable media including program instruc- tion       over the Internet’’ invalid under § 101 based tions’ to an otherwise non-statutory process claim is in- upon the court’s interpretation of Bilski 
sufficient to make it statutory ’’46 Consequently, this Concerning the method claim, the court considered claim also failed the Bilski test 
both the ‘‘transformation’’ and ‘‘machine’’ prongs of the In at least one instance, the U S  International Trade Bilski test  In concluding that there was no transforma- Commission has interpreted the ‘‘machine’’ prong of tion, the court focused on the intangibility of the ma- Bilski less stringently than did the district courts in the nipulated data  According to the court, transformation cases discussed above  In In the Matter of Certain Video is limited to transformation of a physical article or sub- Game Machines and Related Three-Dimensional Point- stance  Accordingly, the method claim did not qualify ing Devices,47 the accused infringer filed a motion for because the data representing credit cards did not rep- summary judgment alleging that the asserted claims resent tangible articles but instead an intangible series impermissibly sought to patent a mathematical algo- of rights and obligations existing between the account rithm  According to the movant, the recitations of a ‘‘3D pointing device,’’ ‘‘handheld device,’’ or ‘‘free space Concerning whether the claimed method was tied to pointing device’’ were not sufficient to tie the claims to a particular machine, the court assessed whether ‘‘reci- a particular machine, but served ‘‘only to limit the field-of-use of the claimed mathematical algorithm and [did] not otherwise impart patentability on the claimed math- Id  at *3  The court relied upon the holdings in Ex parte Gutta, No  2008-3000 at 5-6 (B P A I  Jan  15, 2009) (stating In denying the motion for summary judgment, the ‘‘[t]he recitation in the preamble of ‘[a] computerized method ITC first noted that, ‘‘[w]hile the ultimate determination performed by a data processor’ adds nothing more than a gen- of whether the asserted claims are patentable under eral purpose computer that is associated with the steps of the § 101 is a question of law, the Federal Circuit has ac- process in an unspecified manner ’’); Ex parte Nawathe, No 
2007-3360, 2009 WL 327520, *4 (B P A I  Feb  9, 2009) (finding‘‘the computerized recitation purports to a general purpose processor [], as opposed to a particular computer specifically programmed for executing the steps of the claimed method ’’); and Ex parte Cornea-Hasegan, No  2008-4742 at 9-10 (B P A I 
Jan  13, 2009) (indicating the appellant does not dispute ‘‘the recitation of a processor does not limit the process steps to any 44 Claims having this format are called ‘‘Beauregard’’ specific machine or apparatus ’’)  The court also cited Cyber- claims and were found to not be barred by the traditional source Corp  v  Retail Decisions Inc , (discussed below), in sup- printed matter rule in In re Beauregard, 53 F 3d 1583, 1584, 35 port of its interpretation of the required ‘‘particular machine ’’ 37 620 F  Supp  2d 1068, 92 USPQ2d 1011 (N D  Cal  2009) 47 2009 WL 1070801 (U S I T C  2009) 
knowledged that ‘there may be cases in which the legal given a dataset of feature vectors associated with the question as to patentable subject matter may turn on subsidiary factual issues’ ’’ (citation omitted)  In con- for each binary partition under consideration, rank- struing the claims, the tribunal found that there was a ing features using two-category feature ranking; and genuine dispute as to whether the claimed ‘‘devices’’represented a ‘‘particular machine’’ under the Bilski while the predetermined number of features has not test and whether the claimed ‘‘two-dimensional rota- yet been selected: picking a binary partition p; tional transform’’ was merely a mathematical calcula- selecting a feature based on the ranking for binary tion or instead meant ‘‘changing the mathematical rep- resentation of a two-dimensional quantity from oneframe of reference to a differently-oriented frame of ref- adding the selected feature to an output list if not al- erence’’ as asserted by the patentee  Additionally, the ready present in the output list and removing the se- dispute over the meaning of the claimed ‘‘two- lected feature from further consideration for the bi- dimensional rotational transform’’ also raised a dis- puted issue as to whether this element recited a trans- Notably, while the independent claim failed the formation that would qualify under the ‘‘transforma- machine-or-transformation test, its dependent claim tion’’ prong of Bilski  Given these disputed issues, the was eligible because it recited, ‘‘further comprising us- ITC concluded that it was inappropriate to grant sum- ing the selected features in training a classifier for clas- mary judgment as to the patent eligibility of the claims 
sifying data into categories ’’ In view of the specifica- A similar conclusion was reached in Versata Soft- tion, the board indicated that the ‘‘classifier’’ was a par- ware Inc  v  Sun Microsystems Inc ,48 in which the dis- ticular machine ‘‘in that it performs a particular data trict court denied the defendant’s motion for summary classification function that is beyond mere general pur- judgment of invalidity under Section 101 based upon pose computing ’’53 The board also concluded that the the Bilski court’s refusal ‘‘to adopt a broad exclusion claim ‘‘transforms a particular article into a different over software or any other such category of subject state or thing, namely by transforming an untrained matter beyond the exclusion of claims drawn to funda- classifier into a trained classifier ’’54 In Ex parte Casati,55 the board reversed the examin- Less stringent ‘‘machine’’ prong analyses are also er’s Section 101 rejection of a method claim reciting: found at the board level  For example, in Ex parteSchrader,50 the board held patent-eligible under Bilski A method of analyzing data and making predictions, reading process execution data from logs for a busi- A method for obtaining feedback from consumers re- ceiving an advertisement from an ad provided by anad provider through an interactive channel, the collecting the process execution data and storing the process execution data in a memory defining a ware-house; creating a feedback panel including at least one feed-back response concerning said advertisement; and analyzing the process execution data; generatingprediction models in response to the analyzing; and providing said feedback panel to said consumers, using the prediction models to predict an occurrence said feedback panel being activated by a consumer to of an exception in the business process 
provide said feedback response concerning said ad-vertisement to said ad provider through said interac- In this case, giving consideration to the specification, which ‘‘unequivocally describes the data warehouse aspart of the overall system apparatus, and subsequent Here, the board found ‘‘interactive channel’’ to be descriptions describe the memory/warehouse device in part of an ‘‘overall patent eligible system of appara- terms of machine executable functions,’’ the board con- tuses’’ when viewed in the context of the specification, cluded that ‘‘one of ordinary skill in the art would un- which included ‘‘the Internet and World Wide Web, In- derstand that the claimed storing of process execution teractive Television, and self service devices, such as In- data in a memory defining a warehouse constitutes formation Kiosks and Automated Teller Machines ’’51 patent-eligible subject matter under § 101 because the In another recent decision, Ex parte Forman,52 the memory/warehouse element ties the claims to a particu- board found a ‘‘computer-implemented feature selec- tion method’’ including a ‘‘classifier’’ eligible under Other recent board decisions have reached the oppo- Section 101 because it satisfied both the machine and transformation prong  Here, the ‘‘classifier’’ was recitedin a dependent claim, in which its independent claim re-cited: 53 Id  at 13 
54 Id  See also Ex parte Busche, No  2008-004750 (B P A I 
A computer-implemented feature selection method May 28, 2009) (holding a process claim and a computer pro- for selecting a predetermined number of features for gram product claim, each reciting training a machine, ‘‘are di- a set of binary partitions over a set of categories rected to machines that have such structure as may be adaptedby training ’’) 55 No  2009-005786 (B P A I  July 31, 2009) 
48 2009 WL 1084412, *1 (E D  Tex  March 31, 2009) 
56 Id  at 7  See also Ex parte Dickerson, No  2009-001172 at 49 Citing Bilski, 545 F 3d at 959 n  23 
16 (B P A I  July 9, 2009) (holding claims that ‘‘recite a comput- 50 No  2009-009098 (B P A I  Aug  31, 2009) 
erized method which includes a step of outputting information from a computer       are tied to a particular machine or appa- 52 No  2008-005348 (B P A I  Aug  17, 2009) 
PATENT, TRADEMARK & COPYRIGHT JOURNAL implemented methods ineligible under the Bilski test transformation test applied to this type of claim 63 because the claims failed to tie the method steps to any Then, applying the Bilski test, the board concluded that concrete parts, devices, or combinations of devices  For the claim did not qualify  According to the board, the example, in Ex parte Holtz,57 the board found ineligible under Section 101 a ‘‘method for comparing file tree de-scriptions’’ because the claim ‘‘obtains data (a file struc- does not transform physical subject matter and is not ture), compares data (file structures), generates a tied to a particular machine        Limiting the claims change log, and optimizes the change log without tying to computer readable media does not add any practi- these steps to any concrete parts, devices, or combina- cal limitation to the scope of the claim  Such a field- tions of devices’’ and the ‘‘file structures’’ did not repre- of-use limitation is insufficient to render an other- Similarly, in Ex parte Gutta,58 the board held ineli- gible under § 101 a ‘‘method for identifying one or moremean items for a plurality of items       having a sym- II  The Current Scope of Patent Eligibility bolic value of a symbolic attribute,’’ concluding that the These recent cases establish that some types of meth- claim ‘‘computes a variance and selects a mean item ods are clearly patent-eligible under Section 101, others without tying these steps to any concrete parts, devices, clearly are not eligible, and yet others may be depend- or combinations of devices’’ and ‘‘symbolic values are ing on how they are described and claimed 
neither physical objects nor do they represent physicalobjects ’’ First, the eligibility of system and apparatus claims is largely unaffected by the Bilski decision, with the ca- In contrast to the district court’s decision in Cyber- veat that such claims may be more closely scrutinized source Corp , discussed supra, in a recent board deci- for compliance with Diamond v  Diehr and Gottschalk sion, Ex parte Bodin,59 ‘‘a computer program product’’ v  Benson, which prohibit patenting of a claim directed was found to be patent-eligible subject matter as being to ‘‘laws of nature, natural phenomena, [or] abstract embodied in a ‘‘computer readable medium ’’ Here, the board considered whether the phrase ‘‘recorded on the Also, methods that are performed at least in part by a recording medium’’ as it is recited in the body of the machine qualify for patent eligibility under Section 101 
claims was the same as ‘‘recorded on a computer- Thus, for example, some computer-implemented and readable medium ’’ Acknowledging the differences be- software-related inventions remain patentable as long tween a statutory claim to a data structure stored on a as they are properly described and claimed as being computer readable medium compared to a nonstatutory performed by a computer or computer components 
claim to a data structure that referred to ideas reflected The tie to a machine, however, cannot merely be im- in nonstatutory processes, the board stated: ‘‘[w]hen plicit based upon the description and context of the ap- functional descriptive material is recorded on some plication or general language in the preamble of the computer-readable medium, it becomes structurally claim  Instead, the use of a machine to perform one or and functionally interrelated to the medium and will be more of the claimed functions must be expressly de- statutory in most cases since use of technology permits scribed in the body of the claim so as to be a meaning- the function of the descriptive material to be real- ful limitation on the claim  If a method claim can be read in such a way that all functions can be performed Similarly, in Ex parte Azuma,61 a claim to a ‘‘com- by a human, it will likely not pass the machine prong of puter program product       comprising: a computer us- able medium’’ was found to be directed to statutory The ‘‘Interim Examination Instructions for Evaluat- subject matter under § 101 because the language ‘‘com- ing Subject Matter Eligibility Under 35 U S C  § 101’’ re- puter usable medium’’ referred to tangible storage me- cently issued by the Patent and Trademark Office con- dia, such as a server, floppy drive, main memory and firm that the recitation of a general purpose computer hard disk as disclosed by appellant’s specification, and is sufficient to satisfy Section 101 where the general did not ‘‘implicate the use of a carrier wave ’’ purpose computer is ‘‘programmed to perform the pro- In an older decision, Ex parte Cornea-Hasegan,62 cess steps,       in effect, becom[ing] a special purpose however, the Board seemingly came to the opposite conclusion, holding that a claim reciting ‘‘a computer Concerning data transformation, there seems to be readable media including program instructions which agreement of the Federal Circuit and at least one dis- when executed by a processor cause the processor to trict court that a method that is both limited to transfor- perform’’ a series of steps was not patent-eligible under mation of specific data and limited to a visual depiction Bilski  The board first determined that ‘‘analysis of a representing specific objects or substances qualifies un- ‘manufacture’ claim and a ‘process’ claim is the sameunder 63 Id  at 11 
57 No  2008-004440 at 12-13 (B P A I  Aug  24, 2009) 
65 Diamond v  Diehr, 450 U S  175, 185, 205 USPQ 488 58 No  2008-004366 at 10-11 (B P A I  Aug  10, 2009) 
(1980); Gottschalk v  Benson, 409 U S  63, 67, 175 USPQ 673 59 No  2009-002913 (B P A I  Aug  5, 2009) 
60 Id  at 10 (comparing In re Lowry, 32 F 3d 1579, 1583-84, 66 ‘‘Interim Examination Instructions for Evaluating Sub- 32 USPQ2d 1031 (Fed  Cir  1994) to In re Warmerdam, 33 F 3d ject Matter Eligibility Under 35 U S C  § 101,’’ U S  Patent and 1354, 1361-62, 31 USPQ2d 1754 (Fed  Cir  1994)) 
Trademark Office, Aug  24, 2009, at 6 (78 PTCJ 530, 8/28/09) 
61 No  2009-003902 at 10 (B P A I  Sept  14, 2009) 
The authors’ recent experiences with examiners suggest that 62 No  2008-004742 (B P A I  Jan  13, 2009) 
the examiners are following these instructions 
der Section 101 67 Thus, claims analogous to those in In Concerning claims directed to computer program re Abele68 in which ‘‘data clearly represented physical products, one district court has held that appending ‘‘A and tangible objects, namely the structure of bones, or- computer readable media including program instruc- gans, and other body tissues [so as to recite] the trans- tions’’ to an otherwise non-statutory process claim is in- formation of that raw data into a particular visual depic- sufficient to make it statutory 72 The board has also tion of a physical object on a display’’ are patent- held ineligible claims to ‘‘a computer readable me- dia ’’73 The board has, however, also upheld the eligibil-ity of ‘‘a computer program product’’ as being embod- ied in a computer readable medium 74 Given these in- Bilski has had a significant impact in eliminating consistent decisions, the patent eligibility of claims in patent protection for inventions that are performed en- tirely by humans or can be interpreted as such if read Concerning claims directed to generalized computer broadly  This includes claims that describe processes processing functions, several Board decisions suggest for creating or manipulating legal and financial docu- that, absent a tie to a concrete real-world application, ments and relationships  In this area in particular, many such claims are likely to be deemed an ‘‘algorithm’’ un- pending applications filed prior to Bilski are no longer der Benson and therefore held to be non-statutory  75 patent-eligible, and many issued patents are no longer Any recitation of a specific field of use for the claimed valid  This retroactive impact of the Bilski decision is process or use of the outcome of such processes are troubling, given the investment in these patents and ap- also more likely to be found ‘‘field-of-use’’ or ‘‘post- plications, which have now been rendered essentially solution activity’’ limitations insufficient to render the worthless despite the suggestion in the Federal Circuit’s claim patent-eligible  Thus, the more tied a claimed pro- earlier State Street decision, now overruled, that such cess is to tangible results or particular applications (not claims qualified for patent protection 
just fields of use), the more likely it is to qualify under Inventions that do not fit within the four statutory categories are also not patent-eligible  The Federal Cir-cuit and the board have rejected claims directed to ‘‘a III  Presenting and Claiming Methods in Patent signal,’’ ‘‘a paradigm,’’ ‘‘a user interface’’ and ‘‘a corr-elator’’ on the basis that these items did not qualify as a ‘‘machine, manufacture, composition of matter or pro- Several strategies for describing and claiming meth- cess’’ under § 101  70 There is also an increasing focus ods or processes in patent applications may avoid or on the tangibility of the claimed invention in that, to minimize potential Section 101 problems 
qualify as a ‘‘machine’’ or ‘‘manufacture’’ under Section First, the description provided in a patent application should include well-defined steps or functions associ-ated with method or process  For example, when the claims include ‘‘initiating’’ method steps, a description Remaining areas of uncertainty concerning the scope of well-defined physical steps or functions for initiating of Section 101 include (1) what qualifies under Bilski as should be provided, and a concrete item, machine, de- a ‘‘transformation of an article or data,’’ (2) whether vice, or component that is responsible for the initiating claims to computer programs (Beauregard claims) function should be identified  For claiming ‘‘identify- qualify, and (3) whether internal computer processing ing’’ method steps, provide specific parameters for functionality not tied to a specific application or tan- making the identification, such as according to a speci- fied measurement 76 Where data is involved, the source Concerning data transformation, other than Abele- and type of data should be specified 
style claims discussed above, what qualifies as a data or Also, drawings should be provided that depict the article transformation remains unclear  Claims that concrete item, device, component or combination have been held not to meet the transformation prong in- thereof, and each method or process step or function clude claims directed to the creation or manipulation of should be linked expressly to at least one item, device data representing an intangible series of rights and ob- or component in the drawings that performs the step or ligations (e g , credit card data) and claims directed to function  Broadening language indicating that other the transformation or manipulation of legal obligations components may also be used to perform the function and relationships  Beyond these specific examples, it is may also be included to avoid an unduly narrow inter- difficult to predict what will or will not qualify as a data or article transformation under Bilski 
The claims should affirmatively claim the device, ma- chine or component performing each step or function 
67 In re Bilski, 545 F 3d at 963; Research Corporation Tech- For computer or software-related inventions, the de- nologies, 2009 WL 2413623 at *9 
scription should specify that the software functionality 68 The claimed process involved graphically displaying vari- ances of data from average values wherein the data was X-rayattenuation data produced in a two dimensional field by a com- 72 Cybersource Corp , 620 F  Supp  2d at 1080 
puted tomography scanner  See In re Bilski, 545 F 3d at 962- 73 Cornea-Hasegan, No  2008-004742 
74 Ex parte Bodin, No  2009-002913 (B P A I  Aug  5, 2009) 
69 In re Bilski, 545 F 3d at 963 
75 E g , Ex parte Greene, No  2008-004073 (B P A I  Apr  24, 70 In re Nuijten 500 F 3d 1346, 1357, 84 USPQ2d 1495 (Fed 
2009); Daughtrey, No  2008-000202; Ex parte Arning, No 
Cir  2007) (74 PTCJ 631, 9/28/07) (signal); In re Ferguson, 558 2008-003008 (B P A I  Mar  30, 2009); Cybersource Corp , 620 F 3d 1359, 1366, 90 USPQ2d 1035 (Fed  Cir  2009) (77 PTCJ F  Supp 2d at 1080 (concerning claim 2) 
489, 3/13/09) (paradigm); Ex parte Daughtrey, No  2008- 76 See Brief of American Bar Association as Amicus Curiae 000202 (B P A I  Apr  8, 2009) (user interface); Ex parte Laba- Supporting Respondent, Bilski v  Kappos, No  08-964, ABA die, No  2008-004310 (B P A I  May 6, 2009) (correlator) 
Amicus Br  at 12-13 (U S  amicus brief filed Oct  2, 2009) (78 71 E g , Nuijten, 500 F 3d at 1356-7 
PATENT, TRADEMARK & COPYRIGHT JOURNAL is performed by a computer or computer components 
patent or published application, the option of importing Specificity as to the type of computer component per- subject matter into the specification is limited to ‘‘non- forming each function may be helpful in establishing essential’’ subject matter  In other words, the specifica- eligibility under the Bilski test 
tion can only be amended to disclose a machine for per-forming process steps as long as one skilled in the art IV  Fixing Pre-Bilski Applications to Meet the New would recognize from the original disclosure that the process is implemented by a machine  The key in mak- For patent applications filed prior to the Bilski deci- ing this type of amendment is avoiding (or overcoming) sion, it can be challenging to meet the new require- a rejection under 35 U S C  § 112, para  1, for lack of ments for patent eligibility, particularly when no ma- chine or transformations were expressly described in If incorporation by reference is not an option, a patent applicant may submit evidence, such as a decla- In some cases, there may be sufficient explicit de- ration by the inventor or a duly qualified technical ex- scription of a machine, e g , a computer, such that the pert, demonstrating that one skilled in the art would un- machine can be added into the body of the claims  For derstand the disclosed method to be one performed by example, patent applications for computer-related in- a machine  Unlike attorney argument, which can be dis- ventions sometimes contain a generic description of regarded, such evidence must be considered by the ex- computers that are used to perform the claimed method, and such a generic description may be suffi- One other option is to reformat the claims  Since Bil- cient to impart patent eligibility to the claims when the ski ostensibly does not apply to system and apparatus general-purpose computer is programmed to become a claims, in some instances it may be possible for an ap- plicant to convert his method claims into system claims For patent applications lacking in an explicit descrip- to avoid application of the Bilski test  This strategy, tion of any machine, however, the application may in- however, is unlikely to succeed where the patent speci- corporate by reference patents or publications that can fication does not describe such a system for implement- be used to bolster the specification and provide support ing the method and therefore does not provide the req- for the requisite claim amendments  When an applica- uisite disclosure of the claimed invention under Section tion incorporates by reference a U S  patent or pub- lished U S  patent application, any description from the incorporated references, whether or not the subject The future of the Bilski machine-or-transformation matter is ‘‘essential’’ to support the claims, may be im- test now rests with the Supreme Court  Regardless of ported into the specification  This option may enable the outcome of the appeal, however, it is clear that the importation of the requisite description of a machine, scope of statutory subject matter under Section 101 has which can then also be recited in the claims 77 When been narrowed  The Supreme Court now has a chance the document incorporated by reference is not a U S 
to clarify what has been excluded; it may even reject ormodify the Bilski machine-or-transformation test  How 77 Manual of Patent Examining Procedure, Eighth Ed , Rev 
this will affect the development and protection of cur- 7/2008, at § 608 01(P); see also 37 C F R  § 1 57 
rent and future technologies remains to be seen 
Source: http://www dorsey com/files/upload/luedke_bna_patent_journal_nov09 pdf
(resolução 404 2012 retificação 19062012)
RESOLUÇÃO Nº 404 , DE 12 DE JUNHO DE 2012 Dispõe sobre padronização dos procedimentos administrativos na lavratura de Auto de Infração, na expedição de notificação de autuação e de notificação de penalidade de multa e de advertência, por infração de responsabilidade de proprietário e de condutor de veículo e da identificação de condutor infrator, e dá outras providências 
Cheloidi e cicatrici ipertrofiche in dermatologia
a cura del dr  Antonio Del Sorbo - Specialista in Dermatologia e Venereologia antoniodelsorbo@libero it I Cheloidi di Alibert A volte una ferita anche apparentemente banale, guarisce lasciando una cicatrice voluminosa, rossastra e soprattutto antiestetica  I cheloidi sono cicatrici abnormi che possono far seguito a intervento chirurgico (es: tiroide, mammella, etc) e questo u
Context 2: Brooksley Elizabeth Born (born August 27, 1940) is an American attorney and former public official who, from August 26, 1996, to June 1, 1999, was chair of the Commodity Futures Trading Commission (CFTC), the federal agency which oversees the U S  futures and commodity options markets  During her tenure on the CFTC, Born lobbied Congress and the President to give the CFTC oversight of off-exchange markets for derivatives, in addition to its role with respect to exchange-traded derivatives, but her warnings were ignored or dismissed, and her calls for reform resisted by other regulators <ref name="nytimes">Goodman, Peter S  The Reckoning - Taking Hard New Look at a Greenspan Legacy, The New York Times, October 9, 2008 </ref> Born resigned as chairperson on June 1, 1999, shortly after Congress passed legislation prohibiting her agency from regulating derivatives 

In 2009, Born received the John F  Kennedy Profiles in Courage Award, along with Sheila Bair of the Federal Deposit Insurance Corporation, in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis 

Early life and education
Born graduated from Abraham Lincoln High School (San Francisco, California) at the age of 16  She then attended Stanford University, where she majored in English and was graduated with the class of 1961   She initially wanted to become a doctor, but a guidance counsellor at Stanford advised her against medicine, so she majored in English literature instead 

She then attended Stanford Law School, one of only seven women in her class   She was the first female student ever to be named president of the Stanford Law Review  She received the "Outstanding Senior" award and graduated as valedictorian of the class of 1964 

Legal career
Immediately after law school Born was selected as a law clerk to judge Henry Edgerton of the U S  Court of Appeals for the District of Columbia Circuit  It was during this time that she met her first husband, Jacob C  Landau, who was a journalist covering the Federal courts at the time  Following her clerkship, she became an associate at the Washington, D C -based international law firm of Arnold & Porter  Born was attracted to Arnold & Porter because it was one of the few major law firms to have a woman partner at that time, Carolyn Agger, who was the head of the tax practice  Born took a two-year leave of absence from Arnold & Porter to accompany her first husband to Boston, where he had received a fellowship  During that time she worked as a research assistant to law professor Alan Dershowitz 

Born's early career at Arnold & Porter focused on international trade law, in which she represented a number of Swiss industries and the government of Switzerland  She developed a practice representing clients in numerous complex litigation and arbitration cases involving financial market transactions  Among her high-profile cases was the matter of the Hunt Brothers attempt to corner the market in silver in the 1970s  She made partner at Arnold & Porter, after moving to a three-day schedule to help raise her second child, and eventually rose to be the head of the firm's derivatives practice 

Born was among the first female attorneys to systematically address inequities regarding how the laws treated women  Born and another female lawyer, Marna Tucker, taught what is considered to have been the first "Women and the Law" course at Catholic University’s Columbus School of Law  The class exclusively concerned prejudicial treatment of women under the laws of the United States, past and present   Born and Tucker were surprised to discover that there was no textbook on the issue at the time   Born is also one of the co-founders of the National Women's Law Center  Born also helped rewrite the American Bar Association rules to make it possible for more women and minorities to sit on federal bench 

During her long legal career, and into her retirement, Born did much pro bono and other types of volunteer work  She was active in the American Bar Association, the largest professional organization of lawyers in the United States   Initially Born was named a member of the governing council of the ABA's Individual Rights Section, eventually becoming chairperson   Born and Tucker founded the ABA Women's Caucus, the first organization of female lawyers in the ABA   She held several other senior positions in the ABA, including being named the first woman member of the ABA's Standing Committee on the Federal Judiciary   As a member of the Judiciary Committee, Born provided testimony and opinion on persons nominated for federal judgeships  In 1980 she was named chair of the committee   As chair of the committee, Born was invited to address the U S  Congress regarding the nomination of Judge Sandra Day O'Connor to the U S  Supreme Court 

In 1993, Born's name was floated as a possible candidate for Attorney General of the United States, but Janet Reno was nominated 

In July 2009, Nancy Pelosi appointed Brooksley Born as a commissioner to the Financial Crisis Inquiry Commission (FCIC) 

Born and the OTC derivatives market
Born was appointed to the CFTC on April 15, 1994, by President Bill Clinton  Due to litigation against Bankers Trust Company by Procter and Gamble and other corporate clients, Born and her team at the CFTC sought comments on the regulation of over-the-counter derivatives, a first step in the process of writing CFTC regulations to supplement the existing regulations of the Federal Reserve System,  the Options Clearing Corporation, and the National Association of Insurance Commissioners  Born was particularly concerned about swaps, financial instruments that are traded over the counter between banks, insurance companies or other funds or companies, and thus have no transparency except to the two counterparties and the counterparties' regulators, if any   CFTC regulation was strenuously opposed by Federal Reserve chairman Alan Greenspan, and by Treasury Secretaries Robert Rubin and Lawrence Summers



ground_truth: It required the transformation to be limited to specific data and a visual depiction representing specific objects or substances.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Who was Brooksley Elizabeth's first husband?

Read the following text and answer briefly.

Context 2:  On May 7, 1998, former SEC Chairman Arthur Levitt joined Rubin and Greenspan in objecting to the issuance of the CFTC's concept release  Their response dismissed Born's analysis and focused on the hypothetical possibility that CFTC regulation of swaps and other OTC derivative instruments could create a "legal uncertainty" regarding such financial instruments,  hypothetically reducing the value of the instruments  They argued that the imposition of regulatory costs would "stifle financial innovation" and encourage financial capital to transfer its  transactions offshore  The disagreement between Born and the Executive Office's top economic policy advisors has been described not only as a classic Washington turf war, but also a war of ideologies,  insofar as it is possible to argue that Born's actions were consistent with Keynesian and neoclassical economics while Greenspan, Rubin, Levitt, and Summers consistently espoused neoliberal, and neoconservative policies 

In 1998, a trillion-dollar hedge fund called Long Term Capital Management (LTCM) was near collapse   Using mathematical models to calculate debt risk, LTCM used derivatives to leverage $5 billion into more than $1 trillion, doing business with fifteen of Wall Street's largest financial institutions   The derivative transactions were not regulated, nor were investors able to evaluate LTCM's exposures   Born stated, "I thought that LTCM was exactly what I had been worried about"   In the last weekend of September 1998, the President's working group was told that the entire American economy hung in the balance   After intervention by the Federal Reserve, the crisis was averted   In congressional hearings into the crisis, Greenspan acknowledged that language had been introduced into an agriculture bill that would prevent CFTC from regulating the derivatives which were at the center of the crisis that threatened the US economy   U S  Representative Maurice Hinchey (D-NY) asked "How many more failures do you think we'd have to have before some regulation in this area might be appropriate?"  In response, Greenspan brushed aside the substance of Born's warnings with the simple assertion that "the degree of supervision of regulation of the over-the-counter derivatives market is quite adequate to maintain a degree of stability in the system"  Born's warning was that there wasn't any regulation of them   Born's chief of staff, Michael Greenberger summed up Greenspan's position this way: "Greenspan didn't believe that fraud was something that needed to be enforced, and he assumed she probably did  And of course, she did "  Under heavy pressure from the financial lobby, legislation prohibiting regulation of derivatives by Born's agency was passed by the Congress   Born resigned on June 1, 1999 

The derivatives market continued to grow yearly throughout both terms of George W  Bush's administration  On September 15, 2008, the bankruptcy of Lehman Brothers forced a broad recognition of a financial crisis in both the US and world capital markets   As Lehman Brothers' failure temporarily reduced financial capital's confidence, a number of newspaper articles and television programs suggested that the failure's possible causes included the conflict between the CFTC and the other regulators Faiola, Anthony, Nakashima, Ellen and Drew, Jill  The Crash: Risk and Regulation - What Went Wrong, The Washington Post, October 15, 2008 

Born declined to publicly comment on the unfolding 2008 crisis until March 2009, when she said: "The market grew so enormously, with so little oversight and regulation, that it made the financial crisis much deeper and more pervasive than it otherwise would have been " She also lamented the influence of Wall Street lobbyists on the process and the refusal of regulators to discuss even modest reforms 

An October 2009 Frontline documentary titled "The Warning"  described Born's thwarted efforts to regulate and bring transparency to the derivatives market, and the continuing opposition thereto  The program concluded with an excerpted interview with Born sounding another warning: "I think we will have continuing danger from these markets and that we will have repeats of the financial crisis -- may differ in details but there will be significant financial downturns and disasters attributed to this regulatory gap, over and over, until we learn from experience "

In 2009 Born, along with Sheila Bair of the FDIC, was awarded the John F  Kennedy Profiles in Courage Award in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis   According to Caroline Kennedy, "Brooksley Born recognized that the financial security of all Americans was being put at risk by the greed, negligence and opposition of  powerful and well connected interests     The catastrophic financial events of recent months have  proved them [Born and Sheila Bair] right "  One member of the President's working group had a change of heart about Brooksley Born   SEC Chairman Arthur Levitt stated "I've come to know her as one of the most capable, dedicated, intelligent and committed public servants that I have ever come to know", adding that "I could have done much better  I could have made a difference" in response to her warnings 

In 2010, a documentary film Inside Job further alleged that derivatives regulation was ineffective from the Clinton administration on  Along with fellow whistleblower, former IMF Chief Economist Raghuram Rajan, who was also scorned by the economic establishment, Brooksley Born was cited as one of the authorities arguing that financial derivatives increase economic risk 

 Personal life 
Born is married to Alexander E  Bennett (also retired from Arnold & Porter)   She has five adult children - two from a previous marriage to Jacob Landau and three stepchildren  Notably, Born was named a partner at Arnold & Porter while working part-time so she could raise her two young children   When both of her children were school-age, Born returned to practice full-time 

References

External links
Attorney profile at Arnold & Porter
Brooksley Born (2009 Winner) of the Profiles in Courage Award, with acceptance speech transcript and NECN video

Profile at MarketsWiki
Speeches and statements
"Testimony Of Brooksley Born Chairperson of the CFTC Concerning The Over-The-Counter Derivatives Market", before the House Committee On Banking And Financial Services, July 24, 1998 
"The Lessons of Long Term Capital Management L P ", Remarks of Brooksley Born, Chairperson of the CFTC, Chicago-Kent-IIT Commodities Law Institute, Chicago, Illinois, October 15, 1998 
 Interview: Brooksley Born for "PBS Frontline: The Warning", PBS, (streaming VIDEO 1 hour), October 20, 2009 
Articles
Manuel Roig-Franzia  "Credit Crisis Cassandra:Brooksley Born's Unheeded Warning Is a Rueful Echo 10 Years On", The Washington Post, May 26, 2009
 Taibbi, Matt  "The Great American Bubble Machine", Rolling Stone'', July 9–23, 2009

1940 births
American women lawyers
Arnold & Porter people
Clinton administration personnel
Columbus School of Law faculty
Commodity Futures Trading Commission personnel
Heads of United States federal agencies
Lawyers from San Francisco
Living people
Stanford Law School alumni
21st-century American women
Stanford University alumni 
Context 3: \section{Introduction}

Derivate is one of the most important topics not only in mathematics, but also in physics, chemistry, economics and engineering  Every standard Calculus course provides a variety of exercises for the students to learn how to apply the concept of derivative  The types of problems range from finding an equation of the tangent line to the application of differentials and advanced curve sketching  Usually, these exercises heavily rely on such differentiation techniques as Product, Quotient and Chain Rules, Implicit and Logarithmic Differentiation \cite{Stewart2012}  The definition of the derivative is hardly ever applied after the first few classes and its use is not much motivated 

Like many other topics in undergraduate mathematics, derivative gave rise to many misconceptions \cite{Muzangwa2012}, \cite{Gur2007}, \cite{Li2006}  Just when the students seem to learn how to use the differentiation rules for most essential functions, the application of the derivative brings new issues  A common students' error of determining the domain of the derivative from its formula is discussed in \cite{Rivera2013} and some interesting examples of the derivatives, defined at the points where the functions themselves are undefined, are provided  However, the hunt for misconceptions takes another twist for the derivatives undefined at the points where the functions are in fact defined 

The expression of the derivative of the function obtained using differentiation techniques does not necessarily contain the information about the existence or the value of the derivative at the points, where the expression for the derivative is undefined  In this article we discuss a type of continuous functions that have the expression for the derivative undefined at a certain point, while the derivative itself at that point exists  We show, how relying on the formula for the derivative for finding the horizontal tangent line of a function, leads to a false conclusion and consequently to missing a solution  We also provide a simple methodological treatment of similar functions suitable for the classroom 

\section{Calculating the Derivative}

In order to illustrate how deceitful the expression of the derivative can be to a students' eye, let us consider the following problem 

\vspace{12pt}

\fbox{\begin{minipage}{5 25in}

\begin{center}

\begin{minipage}{5 0in}

\vspace{10pt}

\emph{Problem}

\vspace{10pt}

Differentiate the function $f\left(x\right)=\sqrt[3]{x}\sin{\left(x^2\right)}$  For which values of $x$ from the interval $\left[-1,1\right]$ does the graph of $f\left(x\right)$ have a horizontal tangent?

\vspace{10pt}

\end{minipage}

\end{center}

\end{minipage}}

\vspace{12pt}

Problems with similar formulations can be found in many Calculus books \cite{Stewart2012}, \cite{Larson2010}, \cite{Thomas2009}  Following the common procedure, let us find the expression for the derivative of the function $f\left(x\right)$ applying the Product Rule:
\begin{eqnarray}
f'\left(x\right) &=& \left(\sqrt[3]{x}\right)'\sin{\left(x^2\right)}+\left(\sin{\left(x^2\right)}\right)'\sqrt[3]{x} \notag \\ &=& \frac{1}{3\sqrt[3]{x^2}}\sin{\left(x^2\right)}+2x\cos{\left(x^2\right)}\sqrt[3]{x} \notag \\ &=& \frac{6x^2\cos{x^2}+\sin{x^2}}{3\sqrt[3]{x^2}} \label{DerivativeExpression}
\end{eqnarray}

Similar to \cite{Stewart2012}, we find the values of $x$ where the derivative $f'\left(x\right)$ is equal to zero:
\begin{equation}
6x^2\cos{x^2}+\sin{x^2} = 0 
\label{DerivativeEqualZero}
\end{equation}

Since the expression for the derivative (\ref{DerivativeExpression}) is not defined at $x=0$, it is not hard to see that for all values of $x$ from $\left[-1,1\right]$ distinct from zero, the left-hand side of (\ref{DerivativeEqualZero}) is always positive  Hence, we conclude that the function $f\left(x\right)$ does not have horizontal tangent lines on the interval $\left[-1,1\right]$ 

However, a closer look at the graph of the function $f\left(x\right)$ seems to point at a different result: there is a horizontal tangent at $x=0$ (see Figure \ref{fig:FunctionGraph})  

First, note that the function $f\left(x\right)$ is defined in $x=0$  In order to verify if it has a horizontal tangent at this point, let us find the derivative of the function $f\left(x\right)$ using definition:
\begin{eqnarray}
f'\left(0\right) &=& \lim_{h\rightarrow0}{\frac{f\left(0+h\right)-f\left(0\right)}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{h}\sin{\left(h^2\right)}}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\left(\sqrt[3]{h} \cdot {h} \cdot \frac{\sin{\left(h^2\right)}}{h^2}\right)} \notag \\
&=& \lim_{h\rightarrow0}{\sqrt[3]{h}} \cdot \lim_{h\rightarrow0}{h} \cdot \lim_{h\rightarrow0}{\frac{\sin{\left(h^2\right)}}{h^2}} \notag \\
&=& 0 \cdot 0 \cdot 1 = 0 \notag
\end{eqnarray}
since each of the limits above exists  We see that, indeed, the function $f\left(x\right)$ possesses a horizontal tangent line at the point $x=0$ 

\section{Closer Look at the Expression for the Derivative}

What is the problem with the standard procedure proposed by many textbooks and repeated in every Calculus class? The explanation lies in the following premise: the expression of the derivative of the function does not contain the information as to whether the function is differentiable or not at the points where it is undefined  As it is pointed out in \cite{Rivera2013}, the domain of the derivative is determined \emph{a priori} and therefore should not be obtained from the formula of the derivative itself 

In the example above the Product Law for derivatives requires the existence of the derivatives of both functions at the point of interest  Since the function $\sqrt[3]{x}$ is not differentiable in zero, the Product Rule cannot be applied  

In order to see what exactly happens when we apply the Product Rule, let us find the expression for the derivative using definition of the derivative:
\begin{eqnarray}
f'\left(x\right) &=& \lim_{h\rightarrow0}{\frac{f\left(x+h\right)-f\left(x\right)}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}\sin{\left(x+h\right)^2}-\sqrt[3]{x}\sin{\left(x^2\right)}}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\left(\sqrt[3]{x+h}-\sqrt[3]{x}\right)}{h}\sin{\left(x^2\right)}} + \notag \\
&& \lim_{h\rightarrow0}{\frac{\left(\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}\right)}{h}\sqrt[3]{x+h}} \notag \\
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}} \cdot \lim_{h\rightarrow0}{\sin{\left(x^2\right)}} +  \notag \\&& \lim_{h\rightarrow0}{\frac{\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}}{h}} \cdot \lim_{h\rightarrow0}{\sqrt[3]{x+h}} \notag \\
&=& \frac{1}{3\sqrt[3]{x^2}} \cdot \sin{\left(x^2\right)}+2x\cos{\left(x^2\right)} \cdot \sqrt[3]{x} \notag 
\end{eqnarray}
which seems to be identical to the expression (\ref{DerivativeExpression}) 

Students are expected to develop a skill of deriving similar results and know how to find the derivative of the function using definition of the derivative only  But how `legal' are the performed operations?

\begin{figure}[H]
\begin{center}
	\includegraphics[width=6 0in]{sin pdf}
	\vspace{ 1in}
	\caption{Graph of the function $g\left(x\right)=\sqrt[3]{x}\cos{\left(x^2\right)}$}
	\label{fig:GFunction}
\end{center}
\end{figure}

Let us consider each of the following limits: 
\begin{eqnarray*}
&& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}} \notag \\
&& \lim_{h\rightarrow0}{\sin{\left(x^2\right)}}\notag \\
&& \lim_{h\rightarrow0}{\frac{\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}}{h}}\notag \\
&& \lim_{h\rightarrow0}{\sqrt[3]{x+h}} 
\end{eqnarray*}
The last three limits exist for all real values of the variable $x$



ground_truth: Jacob C. Landau.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the correct expression for the derivative of the function?

Read the following text and answer briefly.

Context 3:  However, the first limit does not exist when $x=0$  Indeed
\begin{equation*}
\lim_{h\rightarrow0}{\frac{\sqrt[3]{0+h}-\sqrt[3]{0}}{h}} = \lim_{h\rightarrow0}{\frac{1}{\sqrt[3]{h^2}}} = + \infty
\end{equation*}

This implies that the Product and Sum Laws for limits cannot be applied and therefore this step is not justifiable in the case of $x=0$  When the derivation is performed, we automatically assume the conditions, under which the Product Law for limits can be applied, i e  that both limits that are multiplied exist  It is not hard to see that in our case these conditions are actually equivalent to $x\neq0$  This is precisely why, when we wrote out the expression for the derivative (\ref{DerivativeExpression}), it already contained the assumption that it is only true for the values of $x$ that are different from zero 

Note, that in the case of $x=0$ the application of the Product and Sum Laws for limits is not necessary, since the term $\left(\sqrt[3]{x+h}-\sqrt[3]{x}\right)\sin{\left(x^2\right)}$ vanishes 

The correct expression for the derivative of the function $f\left(x\right)$ should be the following:
\begin{equation*}
f'\left(x\right)  = 
\begin{cases} 
\frac{6x^2\cos{\left(x^2\right)}+\sin{\left(x^2\right)}}{3\sqrt[3]{x^2}}, & \mbox{if } x \neq 0 \\ 
0, & \mbox{if } x = 0 
\end{cases}
\end{equation*}

The expression for the derivative of the function provides the correct value of the derivative only for those values of the independent variable, for which the expression is defined; it does not tell anything about the existence or the value of the derivative, where the expression for the derivative is undefined  Indeed, let us consider the function
\begin{equation*}
g\left(x\right) = {\sqrt[3]{x}}\cos{\left(x^2\right)}
\end{equation*}
and its derivative $g'\left(x\right)$ 
\begin{equation*}
g'\left(x\right) = \frac{\cos{\left(x^2\right)}-6x^2\sin{\left(x^2\right)}}{3\sqrt[3]{x^2}}
\end{equation*}

Similar to the previous example, the expression for the derivative is undefined at $x=0$  Nonetheless, it can be shown that $g\left(x\right)$ is not differentiable at $x=0$ (see Figure \ref{fig:GFunction})  Therefore, we provided two visually similar functions: both have the expressions for their derivatives undefined in zero, however, one of these functions possesses a derivative, but the other one does not 

\section{Methodological Remarks}

Unfortunately, there exist many functions similar to the ones discussed above and they can arise in a variety of typical Calculus problems: finding the points where the tangent line is horizontal, finding an equation of the tangent and normal lines to the curve at the given point, the use of differentials and graph sketching  Relying only on the expression of the derivative for determining its value at the undefined points may lead to missing a solution (as in the example discussed above) or to some completely false interpretations (as in the case of curve sketching) 

As it was discussed above, the expression for the derivative does not provide any information on the existence or the value of the derivative, where the expression itself is undefined  Here we present a methodology for the analysis of this type of functions 

Let $f\left(x\right)$ be the function of interest and $f'\left(x\right)$ be the expression of its derivative undefined at some point $x_{0}$  In order to find out if $f\left(x\right)$ is differentiable at $x_{0}$, we suggest to follow a list of steps:

\begin{enumerate}
  \item Check if the function $f\left(x\right)$ itself is defined at the point $x_{0}$  If $f\left(x\right)$ is undefined at $x_{0}$, then it is not differentiable at $x_{0}$  If $f\left(x\right)$ is defined at $x_{0}$, then proceed to next step  
  \item Identify the basic functions that are used in the formula of the function $f\left(x\right)$, that are themselves defined at the point $x_{0}$, but their derivative is not (such as, for example, the root functions) 
	\item Find the derivative of the function $f\left(x\right)$ at the point $x_{0}$ using definition 
\end{enumerate}

The importance of the first step comes from the fact that most students tend to pay little attention to the functions domain analysis when asked to investigate its derivative  Formally, the second step can be skipped, however it will give the students the insight into which part of the function presents a problem and teach them to identify similar cases in the future  the difficulty of accomplishing the third step depends on the form of the function and sometimes can be tedious  Nevertheless, it allows the students to apply the previously obtained skills and encourages the review of the material 

\begin{figure}[H]
\begin{center}
	\includegraphics[width=6 0in]{cos pdf}
	\vspace{ 1in}
	\caption{Graph of the function $g\left(x\right)=\sqrt[3]{x}\cos{\left(x^2\right)}$}
	\label{fig:GFunction}
\end{center}
\end{figure}

\section{Conclusion}

We discussed the misconception, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not at the points, where the expression is undefined  We considered a typical Calculus problem of looking for the horizontal tangent line of a function as an example  We showed how the search for the values that make the expression of the derivative equal zero leads to missing a solution: even though the expression of the derivative is undefined, the function still possesses the derivative at the point  We provided an example of the function that similarly has the expression for the derivative undefined, however the function is not differentiable at the point  We also presented the methodological treatment of such functions by applying the definition of the derivative, which can be used in the classroom 





ground_truth: It depends on the value of x, either 0 or (6x^2cos(x^2)+sin(x^2))/(3(x^2)^(1/3)).