Talk:Leibniz integral rule

message: The "General form" needs an independent proof as the style of proof in the two previous cases differ.
:There is now a proof of the general form, but I noticed it is not very rigorous.  I just corrected an error in derivation so that it is now completely correct.  Some issues that possibly merit further discussion: the limits of integration (a and b) should be written explicitly to depend on  \Delta\alpha ; the invocation of the MVT on the integral needs further explication (it assumes some continuity results, etc); the statement involving the terms inside the integral (in the limit as  \Delta\alpha \rightarrow 0) involves the interchanging of two limiting processes (differentiation and integration) and hence is only valid in certain cases. Clayt85 (talk) 22:11, 2 September 2009 (UTC)
thread_title: General form

message: There appears to be a lot of duplication between these articles.Dfeuer (talk) 18:19, 7 October 2012 (UTC)
thread_title:  Merge with [[Differentiation under the integral sign]]? 

message: I would like to thank everyone who contributed to this page. I have found it very helpful.
The "Proof of basic form" is, I believe, not complete correct. However, it can be fixed easily. 
		
If we knew apriori that the continuous limit as   h \to 0   of the integral existed, the proof shown would work. But showing that the continuous limit of the integral exists is part of what must be proved.
The author proves that the limit exists and has the right value for ONE sequence   \{ 1/n \}  . But in order to show that the continuous limit as   h \to 0   exists and has the correct value, it must be shown that the sequential limit exists and has the correct value for ALL sequences   \left\{ {a_n } \right\}   such that   a_n \to 0  .
Fortunately, the exact same method of proof will work for ANY sequence   \left\{ {a_n } \right\}  
for which   a_n \to 0  . Once you prove that the limit exist and has the correct value for all sequences, you can complete the proof by invoking the theorem that says:
For any real valued function   f   on a closed rectangle in   \mathbb{R}^n    (or, more generally, on any Separable Metric Space):
   \lim_{x \rightarrow a } f(x)   exists and equals L      if and only if 
   \lim_{n \rightarrow \infty } f(a_n)   exist and equals L for EVERY sequence   \left\{ {a_n } \right\}   such that   a_n \to x   as   n \to \infty  . 
I hope you find this comment helpful.
By the way, I looked at the Fubini proof and, while it's very short and elegant, it's not very intuitive. I prefer this proof because it is more "nuts and bolts". Also, I think using Fubini is really overkill for this simple version of the theorem.
Catgod119 (talk) 00:45, 19 January 2013 (UTC)
thread_title:  The "Proof of basic form" is still not correct 

message: The current description makes heavy use of subscript notation for derivatives.  Subscript/superscript notation is highly ambiguous, having different meanings and conventions in different fields.  Chemistry uses subscripts to represent number of occurances of an atom in a molecule, linear algebra uses subscripts to index elements of vectors and matrices, and here we are introducing a 3rd notation for partial derivatives.  Readers of this article will often be highschool students and freshman.  Some of them will be like me and find this notation makes the formulas too hard to understand.  I recommend standardizing on the Liebnitz notation.  — Preceding unsigned comment added by 68.232.116.79 (talk) 19:26, 6 April 2013 (UTC)
thread_title:  Notation 

message: Proof of basic form
Let:
Failed to parse (lexing error): So that, using difference quotients ——— Substitute equation (1) into equation (2), combine the integrals (since the difference of two integrals equals the integral of the difference) and use the fact that 1/h is a constant:  :\begin{align} u'(x) &= \lim_{h \rightarrow 0} \frac{\int_{y_0}^{y_1}f(x + h, y)\,\mathrm{d}y - \int_{y_0}^{y_1}f(x, y)\,\mathrm{d}y}{h} \\ &= \lim_{h \rightarrow 0} \frac{\int_{y_0}^{y_1}\left( f(x + h, y) - f(x,y) \right)\,\mathrm{d}y}{h} \\ &= \lim_{h \rightarrow 0} \int_{y_0}^{y_1} \frac{f(x + h, y) - f(x, y)}{h} \,\mathrm{d}y \end{align}  — Preceding unsigned comment added by 181.177.248.131 (talk) 16:35, 27 April 2015 (UTC)
thread_title:  There is a parsing error 

message: In the last paragraph, the justification for the continuity of the function
 \dfrac{\partial F}{\partial x}(x, y) = \int_{t_1}^{y} \dfrac{\partial f}{\partial x}(x,t) dt 
seems insufficient. It is certainly continuous in variable y, but it is a function of two variables, x and y.  
121.176.253.42 (talk) 23:42, 23 July 2019 (UTC)
thread_title:  "Alternative Proof of General Form with Variable Limits, using the Chain Rule" is incorrect 

message: Shall we split this article into "Leibniz integral rule" and "Integration by differentiating under the integral sign"? Nerd271 (talk) 16:13, 5 April 2020 (UTC)
thread_title:  Splitting proposal 

message: I think they both are same . One eye Triangle (talk) 10:58, 3 May 2020 (UTC)
replies: text: Feynman's technique for evaluating integrals uses the Leibniz rule for differentiating under the integral sign. It is a technique that uses the theorem, but is not the theorem itself. As an analogy, think of solving a first-order linear ordinary differential equation using an integration factor. (Hope you have gotten this far in calculus.) The integration factor makes use of the Product Rule, but is not the Product Rule itself. Nerd271 (talk) 15:11, 3 May 2020 (UTC)
thread_title:  What is difference between feynman trick integral and Leibniz rule?? 

message: In the current version of the page, there are enumerated Examples 1, 2, 4, 5, and 6. Should we renumber these to include an Example 3, restore a (possibly) previously-deleted Example 3, or add a new example so the numeration makes sense?
75.181.102.130 (talk) 15:22, 12 June 2021 (UTC)
thread_title:  Whither Example 3? 

message: According to https://planetmath.org/differentiationundertheintegralsign, only (almost everywhere) existence of the derivative is required, not continuity. 2601:547:500:E930:8D0A:C216:B0C4:2C6B (talk)  — Preceding undated comment added 17:14, 16 April 2022 (UTC)
thread_title:  Continuity of the derivative required? 

message: In the opening lines of the wikipedia page, it is stated that the limits of the integration need to satisfy the requirements -\infty < a(x), b(x) < \infty. 
However, in the general statement of the theorem further down the page, these requirements are dropped.
Are these requirements missing? Or were they never required to begin with? If missing, then they need to be added to the general theorem. If they are not required, they shouldn't be stated as required in the first place. Whole Oats (talk) 01:44, 18 April 2023 (UTC)
thread_title:  Requirements on Limits of integration 

message: The information block to the right fully blocks (is that why it is called a block ;-)) the last term in the equation:
“\begin{align}
& \frac{d}{dx} \left (\int_{a(x)}^{b(x)} f(x,t)\,dt \right ) \\
&= f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \,dt
\end{align}
where the partial derivative \tfrac{\partial}{\partial x} indicates that inside the integral, only the variation of f(x, t) with x is considered in taking the derivative.“
I only discovered the blocking because the remark about the partial derivative made no sense to me, since no partial derivative was visible to me.
That must be something similar to what another user tried to mend. That user wrote:
”22:55, 21 February 2023‎ Dolphin51 talk contribs‎  53,046 bytes+38‎   →‎top: In my browser the Info box obscured part of this long equation. I have solved the problem by presenting the equation over two lines. If someone knows of a better way to do it - feel free to go ahead.”
While for Dolphin51 the problem may have been solved, for me it is still there.
I am afraid Wikipedia could do with a more structural solution that prevents visual parts of a page from blocking other parts from view. Or is this a browser problem?
Who would know how to get this problem solved? I have a screen photograph ready that illustrates the problem, but cannot see a way to upload it here.Redav (talk) 00:13, 28 January 2024 (UTC)
replies: text: Safari does not show scroll bars unless "active" by default.  Set in your system preferences in appearance to always show them.  This is not a wiki problem, but a safari problem.  There are scroll bars there, safari just does not show them since they are not pretty. AManWithNoPlan (talk) 00:38, 28 January 2024 (UTC)
replies: text:  ooops.  You are rigtht, that setting is only on desktop.  You can scroll it, but there is no visial clue. AManWithNoPlan (talk) 12:06, 28 January 2024 (UTC)
replies: text:  https://en.wikipedia.org/w/index.php?title=Wikipedia%3AVillage_pump_%28technical%29&diff=1199981416&oldid=1199966707 AManWithNoPlan (talk) 12:11, 28 January 2024 (UTC)
replies: text:  There does not seem to be any solution https://stackoverflow.com/questions/73496949/how-to-make-scrollbar-always-visible AManWithNoPlan (talk) 15:19, 31 January 2024 (UTC)
thread_title:  Structural solution for presentation problem requested: parts of equation invisible in Safari browser on iPad 

message: I’m not sure, but apparently some of the calculations have problems. I could just be dumb however 48JCLTalk 18:43, 30 April 2024 (UTC)
thread_title:  Errors