hcju/proofwikips huggingface dataset:
True Statement is implied by Every Statement/Formulation 2/Proof 2
True Statement is implied by Every Statement/Formulation 2/Proof 2 0 32742 182453 2014-04-07T11:54:46Z Lord Farin 560 Created page with "== Theorem ==  : $\vdash q \implies \left({p \implies q}\right)$   == Proof ==  We apply the Method of Truth Tables.  As can be seen by inspection, the Defi..." wikitext text/x-wiki == Theorem == : $\vdash q \implies \left({p \implies q}\right)$ == Proof ==  We apply the Method of Truth Tables. As can be seen by inspection, the truth value under the main connective, the first instance of $\implies$, is $T$ for each boolean interpretation. $\begin{array}{|ccccc|} \hline q & \implies & ( p & \implies & q ) \\ \hline F & T & T & F & F \\ F & T & F & T & F \\ T & T & T & T & T \\ T & T & F & T & T \\ \hline \end{array}$
Axiom:Peano's Axioms
== Axioms == '''Peano's Axioms''' are a set of properties which can be used to serve as a basis for logical deduction of the properties of the natural numbers.  '''Peano's Axioms''' are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ and $0$ as an element of $\N$. Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$. Historically, the existence of $s$ and the existence of $0$ were considered the first two of '''Peano's Axioms''': {{begin-axiom}} {{axiom | n = \text P 1         | q =          | m = 0 \in P         | t = $0$ is an element of $P$ }} {{axiom | n = \text P 2         | q = \forall n \in P         | m = \map s n \in P         | t = For all $n \in P$, its successor $\map s n$ is also in $P$ }} {{end-axiom}} The other three are as follows:  === Formulation 1 === {{:Axiom:Peano's Axioms/Formulation 1}} === Formulation 2 === {{:Axiom:Peano's Axioms/Formulation 2}} == Terminology == === Successor Mapping === {{:Definition:Successor Mapping}} === Non-Successor Element === {{:Definition:Non-Successor Element}} === Peano Structure === Such a set $P$, together with the successor mapping $s$ and non-successor element $0$ as defined above, is known as a '''Peano structure'''. == Also presented as == Some sources present the axioms in a different order. For example, {{BookReference|Number Theory|1964|J. Hunter}} places the induction axiom $(\text P 5)$ as axiom $(3)$ and moves $(\text P 3)$ and $(\text P 4)$ down to be $(4)$ and $(5)$ respectively. == Also defined as == {{:Axiom:Peano's Axioms/Also defined as}} {{expand|subpage with Peano's original exposition/axioms}} == Also known as == {{:Axiom:Peano's Axioms/Also known as}} == Also see == * Equivalence of Formulations of Peano's Axioms * Minimal Infinite Successor Set forms Peano Structure * Principle of Mathematical Induction for Peano Structure {{LinkToCategory|Peano's Axioms|Peano's axioms}} {{NamedforAxiom|Giuseppe Peano|author2 = Richard Dedekind|cat = Peano|cat2 = Dedekind}} == Historical Note == {{:Axiom:Peano's Axioms/Historical Note}} == Sources == * {{BookReference|Numbers, Sets and Axioms|1982|Alan G. Hamilton|prev = Successor Mapping on Natural Numbers has no Fixed Element|next = Natural Number Multiplication is Commutative}}: $\S 1$: Numbers: $1.1$ Natural Numbers and Integers: Examples $1.1 \ \text {(e)}$
Stirling's Formula/Proof 2/Lemma 1
== Lemma == Let $f \left({x}\right)$ be the real function defined on the open interval $\left({-1 \,.\,.\, 1}\right)$ as: :$f \left({x}\right) := \dfrac 1 {2 x} \ln \left({\dfrac {1 + x} {1 - x} }\right) - 1$ Then: :$\displaystyle f \left({x}\right) = \sum_{k \mathop = 1}^\infty \dfrac {x^{2 n} } {2n + 1}$ == Proof == {{begin-eqn}} {{eqn | l = f \left({x}\right)       | o = :=       | r = \frac 1 {2 x} \ln \left({\frac {1 + x} {1 - x} }\right) - 1       | c = for $\left\vert{x}\right\vert < 1$ }} {{eqn | r = \frac 1 {2 x} \left({\ln \left({1 + x}\right) - \ln \left({1 - x}\right)}\right) - 1       | c = Difference of Logarithms }} {{eqn | r = \frac 1 {2 x} \left({\sum_{k \mathop = 1}^\infty \left({\left({-1}\right)^{k-1} \frac {x^k} k}\right) - \sum_{k \mathop = 1}^\infty \left({-\frac {x^k} k}\right)}\right) - 1       | c = Power Series Expansion for $\ln \left({1 + x}\right)$ }} {{eqn | r = \sum_{k \mathop = 1}^\infty \frac {x^{2n} } {2n + 1}       | c = simplifying }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev=Stirling's Formula/Proof 2|next=Stirling's Formula/Proof 2/Lemma 2}}: $\S 17.2$
Stirling's Formula/Proof 2/Lemma 3
== Lemma == Let $\sequence {d_n}$ be the sequence defined as: :$d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$ Then the sequence: :$\sequence {d_n - \dfrac 1 {12 n} }$ is increasing. == Proof == We have: {{begin-eqn}} {{eqn | l = d_n - d_{n + 1}       | r = \map \ln {n!} - \paren {n + \frac 1 2} \ln n + n       | c =  }} {{eqn | o =        | ro= -       | r = \paren {\map \ln {\paren {n + 1}!} - \paren {n + 1 + \frac 1 2} \map \ln {n + 1} + n + 1}       | c =  }} {{eqn | r = -\map \ln {n + 1} - \paren {n + \frac 1 2} \ln n + \paren {n + \frac 3 2} \map \ln {n + 1} - 1       | c = (as $\map \ln {\paren {n + 1}!} = \map \ln {n + 1} + \map \ln {n!}$) }} {{eqn | r = \paren {n + \frac 1 2} \map \ln {\frac {n + 1} n} - 1       | c = }} {{eqn | n = 1       | r = \frac {2n + 1} 2 \map \ln {\frac {1 + \paren {2 n + 1}^{-1} } {1 - \paren {2 n + 1}^{-1} } } - 1       | c = }} {{end-eqn}} Let: :$\map f x := \dfrac 1 {2 x} \map \ln {\dfrac {1 + x} {1 - x} } - 1$ for $\size x < 1$. Then: {{begin-eqn}} {{eqn | l = \map f x       | r = \sum_{k \mathop = 1}^\infty \frac {x^{2 n} } {2 n + 1}       | c = Lemma 1 }} {{eqn | o = <       | r = \frac {x^2} 3 \sum_{k \mathop = 0}^\infty x^{2 n}       | c =  }} {{eqn | n = 1       | o = <       | r = \frac {x^2} {3 \paren {1 - x^2} }        | c = Sum of Infinite Geometric Sequence }} {{end-eqn}} As $-1 < \dfrac 1 {2 n + 1} < 1$ it can be substituted for $x$ in $(1)$: {{begin-eqn}} {{eqn | l = d_n - d_{n - 1}       | o = \le       | r = \frac 1 {3 \paren {\paren {2 n + 1}^2 - 1} }       | c = simplifying }} {{eqn | r = \frac 1 {12 n} - \frac 1 {12 \paren {n + 1} }       | c =  }} {{eqn | ll= \leadsto       | l = d_n - \frac 1 {12 n}        | o = \le       | r = d_{n-1} - \frac 1 {12 \paren {n + 1} } }} {{end-eqn}} Thus the sequence: :$\sequence {d_n - \dfrac 1 {12 n} }$ is increasing.  == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Stirling's Formula/Proof 2/Lemma 2|next = Definition:Integral Form of Gamma Function}}: $\S 17.2$
Definite Integral of Even Function/Corollary
== Corollary to Definite Integral of Even Function ==  Let $f$ be an even function with a primitive on the open interval $\openint {-a} a$, where $a > 0$. Then the improper integral of $f$ on $\openint {-a} a$ is: :$\displaystyle \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x = 2 \int_0^{\mathop \to a} \map f x \rd x$  == Proof == {{begin-eqn}} {{eqn | l = \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x       | r = \lim_{y \to a} \int_{-y}^y \map f x \rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{eqn | r = \lim_{y \mathop \to a} 2 \int_0^y \map f x \rd x       | c = Definite Integral of Even Function }} {{eqn | r = 2 \lim_{y \mathop \to a} \int_0^y \map f x \rd x       | c = Multiple Rule for Limits of Functions }} {{eqn | r = 2 \int_0^{\mathop \to a} \map f x \rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{end-eqn}}
Negative of Absolute Value/Corollary 2
== Corollary to Negative of Absolute Value == Let $x, y \in \R$ be a real numbers. Let $\size x$ be the absolute value of $x$. Then:  :$\size x \le y \iff -y \le x \le y$ that is: :$\size x \le y \iff \begin {cases} x & \le y \\ -x & \le y \end {cases}$  == Proof == === Necessary Condition === Let $\size x \le y$. If $\size x < y$ then from Corollary 1: :$-y < x < y$ Thus: :$-y \le x \le y$ Otherwise, if $\size x = y$ then either $x = y$ or $-x = y$. Hence the result. {{qed|lemma}} === Sufficient Condition === Let $-y \le x \le y$. If $-y < x < y$ then from Corollary 1: :$\size x < y$ Hence: :$\size x \le y$ Otherwise, if either $-y = x$ or $x = y$ then: :$\size x = y$ Hence the result.
Derivative of Identity Function/Corollary
== Corollary to Derivative of Identity Function ==  :$\map {\dfrac {\d} {\d x} } {c x} = c$ where $c$ is a constant.  == Proof == {{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {c x}       | r = c \frac \d {\d x} x       | c = Derivative of Constant Multiple }} {{eqn | r = c \times 1       | c = Derivative of Identity Function }} {{eqn | r = c       | c =  }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Derivative of Constant|next = Power Rule for Derivatives/Corollary}}: $\S 13$: General Rules of Differentiation: $13.3$
Derivative of Constant Multiple/Real/Corollary
== Corollary to Derivative of Constant Multiple == Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then:  :$\map {\dfrac {\d^n} {\d x^n} } {c \map f x} = c \map {\dfrac {\d^n} {\d x^n} } {\map f x}$  == Proof == {{refactor|Expand this into the formal  structure for induction proofs.}} By induction: the base case is for $n = 1$ and is proved in Derivative of Constant Multiple. Now consider $\map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}$, assuming the induction hypothesis $\map {\dfrac {\d^k} {\d x^k} } {c \map f x} = c \map {\dfrac {\d^k} {\d x^k} } {\map f x}$: {{begin-eqn}} {{eqn | l = \map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}       | r = \map {\dfrac \d {\d x} } {\map {\dfrac {\d^k} {\d x^k} } {c \map f x} }       | c = {{Defof|Higher Derivative}} }} {{eqn | r = \map {\dfrac \d {\d x} } {c \map {\dfrac {\d^k} {\d x^k} } {\map f x} }       | c = Induction Hypothesis }} {{eqn | r = c \map {\dfrac \d {\d x} } {\map {\dfrac {\d^k} {\d x^k} } {\map f x} }       | c = Basis for the Induction }} {{eqn | r = c \map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {\map f x}       | c = {{Defof|Higher Derivative}} }} {{end-eqn}} Hence the result by induction.
Axiom:Axiomatization of 1-Based Natural Numbers
{{rename|Hardly elegant}} == Axioms == The following axioms are intended to capture the behaviour of the ($1$-based) natural numbers $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations of addition $+$ and multiplication $\times$ as they pertain to $\N_{>0}$:  {{begin-axiom}} {{axiom | n = \text A         | q = \exists_1 1 \in \N_{> 0}         | m = a \times 1 = a = 1 \times a }} {{axiom | n = \text B         | q = \forall a, b \in \N_{> 0}         | m = a \times \paren {b + 1} = \paren {a \times b} + a }} {{axiom | n = \text C         | q = \forall a, b \in \N_{> 0}         | m = a + \paren {b + 1} = \paren {a + b} + 1 }} {{axiom | n = \text D         | q = \forall a \in \N_{> 0}, a \ne 1         | m = \exists_1 b \in \N_{> 0}: a = b + 1 }} {{axiom | n = \text E         | q = \forall a, b \in \N_{> 0}         | m = \)Exactly one of these three holds:\( }} {{axiom | m = a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} }} {{axiom | n = \text F         | q = \forall A \subseteq \N_{> 0}         | m = \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} }} {{end-axiom}}  == Note == The above axiom schema specifies the old-fashioned definition of the natural numbers as: :$\text{The set of natural numbers} = \set {1, 2, 3, \ldots}$ as opposed to the more modern approach which defines them as: :$\text{The set of natural numbers} = \set {0, 1, 2, 3, \ldots}$ In order to eliminate confusion, on  the set $\set {1, 2, 3, \ldots}$ will be denoted as $\N_{> 0}$ or $\N_{\ne 0}$ or $\N_{\ge 1}$. When $\N$ is used, $\N = \set {0, 1, 2, 3, \ldots}$ is to be understood. == Sources == * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Definition:Natural Numbers|next = Principle of Mathematical Induction/Predicate/One-Based/Proof 2}}: $\S 2.1$: Definition $2.1$
Primitive of Square of Cosine Function/Corollary
== Corollary to Primitive of Square of Cosine Function ==  :$\displaystyle \int \cos^2 x \rd x = \frac {x + \sin x \cos x} 2 + C$ where $C$ is an arbitrary constant.  == Proof == {{begin-eqn}} {{eqn | l = \int \sin^2 x \rd x       | r = \frac x 2 + \frac {\sin 2 x} 4 + C       | c = Primitive of Square of Cosine Function }} {{eqn | r = \frac x 2 + \frac {2 \sin x \cos x} 4 + C       | c = Double Angle Formula for Sine }} {{eqn | r = \frac {x + \sin x \cos x} 2 + C       | c =  }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Cosine Function|next = Primitive of Product of Secant and Tangent}}: $\S 14$: General Rules of Integration: $14.22$
Primitive of Square of Hyperbolic Sine Function/Corollary
== Corollary to Primitive of Square of Hyperbolic Sine Function ==  :$\displaystyle \int \sinh^2 x \rd x = \frac {\sinh x \cosh x - x} 2 + C$ where $C$ is an arbitrary constant.  == Proof == {{begin-eqn}} {{eqn | l = \int \sinh^2 x \rd x       | r = \frac {\sinh 2 x} 4 - \frac x 2 + C       | c = Primitive of Square of Hyperbolic Sine Function }} {{eqn | r = \frac {2 \sinh x \cosh x} 4 - \frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \frac {\sinh x \cosh x - x} 2 + C       | c = rearranging }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Hyperbolic Sine Function|next = Primitive of Square of Hyperbolic Cosine Function}}: $\S 14$: General Rules of Integration: $14.35$
Primitive of Square of Hyperbolic Cosine Function/Corollary
== Corollary to Primitive of Square of Hyperbolic Cosine Function ==  :$\displaystyle \int \cosh^2 x \rd x = \frac {\sinh x \cosh x + x} 2 + C$ where $C$ is an arbitrary constant.  == Proof == {{begin-eqn}} {{eqn | l = \int \cosh^2 x \rd x       | r = \frac {\sinh 2 x} 4 + \frac x 2 + C       | c = Primitive of Square of Hyperbolic Cosine Function }} {{eqn | r = \frac {2 \sinh x \cosh x} 4 + \frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \frac {\sinh x \cosh x + x} 2 + C       | c = rearranging }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Hyperbolic Cosine Function|next = Primitive of Product of Hyperbolic Secant and Tangent}}: $\S 14$: General Rules of Integration: $14.36$
Derivative of Arcsine Function/Corollary
== Corollary to Derivative of Arcsine Function == Let $a \in \R$ be a constant Let $x \in \R$ be a real number such that $x^2 < a^2$. Let $\map \arcsin {\dfrac x a}$ be the arcsine of $\dfrac x a$. Then:  :$\dfrac {\map \d {\map \arcsin {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {a^2 - x^2} }$  == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arcsin {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {\sqrt {1 - \paren {\frac x a}^2} }       | c = Derivative of Arcsine Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \frac 1 a \frac a {\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = \frac 1 {\sqrt {a^2 - x^2} }       | c =  }} {{end-eqn}}  == Also see == * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arcsec \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Square of Hyperbolic Cosecant Function|next = Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form}}: Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $12$.
Derivative of Arccosine Function/Corollary
== Corollary to Derivative of Arccosine Function == Let $a \in \R$ be a constant Let $x \in \R$ be a real number such that $x^2 < a^2$. Let $\map \arccos {\dfrac x a}$ be the arccosine of $\dfrac x a$. Then:  :$\dfrac {\map \d {\map \arccos {\frac x a} } } {\d x} = \dfrac {-1} {\sqrt {a^2 - x^2} }$  == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arccos {\frac x a} } } {\d x}       | r = \frac 1 a \frac {-1} {\sqrt {1 - \paren {\frac x a}^2} }       | c = Derivative of Arccosine Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-1} {\sqrt {\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \frac 1 a \frac {-a} {\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = \frac {-1} {\sqrt {a^2 - x^2} }       | c =  }} {{end-eqn}}  == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arcsec \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form|next = Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form}}: Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $12$.
Derivative of Arctangent Function/Corollary
== Corollary to Derivative of Arctangent Function == Let $x \in \R$. Let $\map \arctan {\dfrac x a}$ denote the arctangent of $\dfrac x a$. Then:  :$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {a^2 + x^2}$  == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arctan {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {1 + \paren {\frac x a}^2}       | c = Derivative of Arctangent Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\frac {a^2 + x^2} {a^2} }       | c =  }} {{eqn | r = \frac 1 a \frac {a^2} {a^2 + x^2}       | c =  }} {{eqn | r = \frac a {a^2 + x^2}       | c =  }} {{end-eqn}}  == Also defined as == This result can also be reported as: :$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {x^2 + a^2}$ == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arcsec \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form|next = Primitive of Reciprocal of x squared plus a squared/Arctangent Form}}: Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $13$.
Derivative of Arccotangent Function/Corollary
== Corollary to Derivative of Arccotangent Function == Let $x \in \R$. Let $\operatorname{arccot} \left({\dfrac x a}\right)$ be the arccotangent of $\dfrac x a$. Then:  :$\dfrac {\mathrm d \left({\operatorname{arccot} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac {-a} {a^2 + x^2}$  == Proof == {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{arccot} \left({\frac x a}\right)}\right)} {\mathrm d x}       | r = \frac 1 a \frac {-1} {1 + \left({\frac x a}\right)^2}       | c = Derivative of Arccotangent Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-1} {\frac {a^2 + x^2} {a^2} }       | c =  }} {{eqn | r = \frac 1 a \frac {-a^2} {a^2 + x^2}       | c =  }} {{eqn | r = \frac {-a} {a^2 + x^2}       | c =  }} {{end-eqn}}  == Also defined as == This result can also be reported as: :$\dfrac {\mathrm d \left({\operatorname{arccot} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac {-a} {x^2 + a^2}$ == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\operatorname{arcsec} \dfrac x a$ * Derivative of $\operatorname{arccsc} \dfrac x a$
Derivative of Arcsecant Function/Corollary 1
== Corollary to Derivative of Arcsecant Function == Let $x \in \R$. Let $\map \arcsec {\dfrac x a}$ be the arcsecant of $\dfrac x a$. Then:  :$\dfrac {\map \d {\map \arcsec {\frac x a} } } {\d x} = \dfrac a {\size x \sqrt {x^2 - a^2} } = \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $x > a$})  \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $x < -a$}) \\ \end{cases}$  == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arcsec {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {\size {\frac x a} \sqrt {\paren {\frac x a}^2 - 1} }       | c = Derivative of Arcsecant Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\size {\frac x a} \frac {\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \frac 1 a \frac {a^2} {\size x {\sqrt {x^2 - a^2} } }       | c =  }} {{eqn | r = \frac a {\size x {\sqrt {x^2 - a^2} } }       | c =  }} {{end-eqn}} {{qed|lemma}} Similarly: {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arcsec {\frac x a} } } {\d x}       | r = \begin {cases} \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $\dfrac x a > 1$}) \\ \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $\dfrac x a < -1$}) \\ \end{cases}       | c = Derivative of Arcsecant Function <br/>and Derivative of Function of Constant Multiple	 }} {{eqn | r = \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end{cases}       | c = simplifying as above }} {{end-eqn}}  == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arccsc \dfrac x a$
Derivative of Arccosecant Function/Corollary
== Corollary to Derivative of Arccosecant Function == Let $x \in \R$. Let $\operatorname{arccsc} \left({\dfrac x a}\right)$ be the arccosecant of $\dfrac x a$. Then:  :$\dfrac {\mathrm d \left({\operatorname{arccsc} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac {-a} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } } = \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}$  == Proof == {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{arccsc} \left({\frac x a}\right)}\right)} {\mathrm d x}       | r = \frac 1 a \frac {-1} {\left\vert{\frac x a}\right\vert \sqrt {\left({\frac x a}\right)^2 - 1} }       | c = Derivative of Arccosecant Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-1} {\left\vert{\frac x a}\right\vert \frac {\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \frac 1 a \frac {-a^2} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } }       | c =  }} {{eqn | r = \frac {-a} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } }       | c =  }} {{end-eqn}} {{qed|lemma}} Similarly: {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{arccsc} \left({\frac x a}\right)}\right)} {\mathrm d x}       | r = \begin{cases} \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\left({\frac x a}\right)^2 - 1} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\left({\frac x a}\right)^2 - 1} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}       | c = Derivative of Arccosecant Function <br/>and Derivative of Function of Constant Multiple	 }} {{eqn | r = \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}       | c = simplifying as above }} {{end-eqn}}  == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\operatorname{arccot} \dfrac x a$ * Derivative of $\operatorname{arcsec} \dfrac x a$