nyuuzyou/wikis huggingface dataset:
Derivative of Hyperbolic Sine

== Theorem ==
:$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$
where $\sinh$ is the hyperbolic sine and $\cosh$ is the hyperbolic cosine.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Derivative of Hyperbolic Cosine
* Derivative of Hyperbolic Tangent
* Derivative of Hyperbolic Cotangent
* Derivative of Hyperbolic Secant
* Derivative of Hyperbolic Cosecant
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $8$.
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $6.$ Hyperbolic trigonometric functions
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* 
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
* 
Category:Derivative of Hyperbolic Sine Function
Derivative of Hyperbolic Sine Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\sinh u} = \cosh u \dfrac {\d u} {\d x}$
where $\sinh$ is the hyperbolic sine and $\cosh$ is the hyperbolic cosine.
== Proof ==
== Also see ==
* Derivative of Hyperbolic Cosine Function
* Derivative of Hyperbolic Tangent Function
* Derivative of Hyperbolic Cotangent Function
* Derivative of Hyperbolic Secant Function
* Derivative of Hyperbolic Cosecant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.31$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Differentiation Formulas: $31$.
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $19$
Category:Derivatives of Hyperbolic Functions
Category:Hyperbolic Sine Function
Category:Derivative of Hyperbolic Sine Function
Derivative of Hyperbolic Tangent

== Theorem ==
:$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x = \dfrac 1 {\cosh^2 x}$
where $\tanh$ is the hyperbolic tangent, $\sech$ is the hyperbolic secant and $\cosh$ is the hyperbolic cosine.
=== Corollary ===
== Proof ==
== Also see ==
* Derivative of Hyperbolic Sine
* Derivative of Hyperbolic Cosine
* Derivative of Hyperbolic Cotangent
* Derivative of Hyperbolic Secant
* Derivative of Hyperbolic Cosecant
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $10$.
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $6.$ Hyperbolic trigonometric functions
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* 
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
* 
Category:Derivatives of Hyperbolic Functions
Category:Hyperbolic Tangent Function
Derivative of Hyperbolic Tangent/Corollary

== Theorem ==
:$\map {\dfrac \d {\d x} } {\tanh x} = 1 - \tanh^2 x$
where $\tanh$ is the hyperbolic tangent.
== Proof ==
== Sources ==
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $6.$ Hyperbolic trigonometric functions
Category:Hyperbolic Tangent Function
Derivative of Hyperbolic Tangent Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\tanh u} = \sech^2 u \dfrac {\d u} {\d x}$
where $\tanh$ is the hyperbolic tangent and $\sech$ is the hyperbolic secant.
== Proof ==
== Also see ==
* Derivative of Hyperbolic Sine Function
* Derivative of Hyperbolic Cosine Function
* Derivative of Hyperbolic Cotangent Function
* Derivative of Hyperbolic Secant Function
* Derivative of Hyperbolic Cosecant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.33$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Differentiation Formulas: $33$.
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $21$
Category:Derivatives of Hyperbolic Functions
Category:Hyperbolic Tangent Function
Derivative of Identity Function

== Theorem ==
Let $X$ be either set of either the real numbers $\R$ or the complex numbers $\C$.
Let $I_X: X \to X$ be the identity function.
Then:
:$\map {I_X'} x = 1$
where $\map {I_X'} x$ denotes the derivative of $I_X$ $x$.
This can be presented for each of $\R$ and $\C$:
=== Derivative of Identity Function for Real Numbers ===
=== Derivative of Identity Function for Complex Numbers ===
== Corollary ==
== Also presented as ==
Note that this can be written:
:$\map {D_x} x = 1$
Using Leibniz's notation for derivatives it can also be stated as:
:$\dfrac {\d x} {\d x} = 1$
== Sources ==
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivatives
Category:Identity Mappings
Derivative of Identity Function/Complex

== Theorem ==
Let $I_\C: \C \to \C$ be the identity function.
Then:
:$\map {I_\C'} z = 1$
=== Corollary ===
== Proof ==
The identity function is defined as $\forall x \in \C: \map {I_\C} z = z$.
Thus:
Category:Complex Differential Calculus
Category:Identity Mappings
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 ==
== Sources ==
* : $\S 13$: General Rules of Differentiation: $13.3$
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
Category:Derivatives
Category:Identity Mappings
Derivative of Identity Function/Real

== Theorem ==
Let $I_\R: \R \to \R$ be the identity mapping on the real numbers $\R$.
Then:
:$\map {I_\R'} x = 1$
== Proof ==
The identity mapping is defined as:
:$\forall x \in \R: \map {I_\R} x = x$
Thus:
Category:Derivatives
Category:Identity Mappings
Derivative of Infinite Product of Analytic Functions

== Theorem ==
Let $D \subset \C$ be open.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$.
Let the product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.
Then:
:$\ds f' = \sum_{n \mathop = 1}^\infty f_n' \cdot \prod_{\substack {k \mathop = 1 \\ k \mathop \ne n} }^\infty f_k$
and the series converges locally uniformly in $D$.
== Outline of Proof ==
Using Logarithmic Derivative of Infinite Product of Analytic Functions we write $\ds \frac {f'} f = \sum_{n \mathop = 1}^\infty \frac {f_n'} {f_n}$, and we multiply this by $f$.
== Proof ==
By Infinite Product of Analytic Functions is Analytic, $f$ is analytic.
We may suppose none of the $f_n$ is identically zero on any open subset of $D$.
Let $E = D \setminus \set {z \in D: \map f z = 0}$.
By Logarithmic Derivative of Infinite Product of Analytic Functions, $\ds \frac {f'} f = \sum_{n \mathop = 1}^\infty \frac {f_n'} {f_n}$ converges locally uniformly in $E$.
By Linear Combination of Convergent Series:
:$\ds f' = \sum_{n \mathop = 1}^\infty f_n' \cdot \prod_{\substack {k \mathop = 1 \\ k \mathop \ne n}}^\infty f_k$ on $E$
By Uniformly Convergent Sequence Multiplied with Function, the series converges locally uniformly in $E$.
By Uniformly Convergent Sequence on Dense Subset, the series converges locally uniformly in $D$.
Let $g$ denote its limit.
By Uniform Limit of Analytic Functions is Analytic, $g$ is analytic in $D$.
By Uniqueness of Analytic Continuation, $f' = g$.
== Also see ==
* Logarithmic Derivative of Infinite Product of Analytic Functions
Category:Complex Analysis
Category:Infinite Products
Derivative of Inverse Function

== Theorem ==
Let $I = \closedint a b$ and $J = \closedint c d$ be closed real intervals.
Let $I^o = \openint a b$ and $J^o = \openint c d$ be the corresponding open real intervals.
Let $f: I \to J$ be a real function which is continuous on $I$ and differentiable on $I^o$ such that $J = f \sqbrk I$.
Let either:
:$\forall x \in I^o: D \map f x > 0$
or:
:$\forall x \in I^o: D \map f x < 0$
Then:
:$f^{-1}: J \to I$ exists and is continuous on $J$
:$f^{-1}$ is differentiable on $J^o$
:$\forall y \in J^o: D \map {f^{-1} } y = \dfrac 1 {D \map f x}$
== Proof ==
From Derivative of Monotone Function, it follows that $f$ is either:
:strictly increasing on $I$ (if $\forall x \in I^o: D \, \map f x > 0$)
or:
:strictly decreasing on $I$ (if $\forall x \in I^o: D \, \map f x < 0$).
Therefore from Inverse of Strictly Monotone Function‎ it follows that $f^{-1}: J \to I$ exists.
As $f$ is continuous, from Image of Interval by Continuous Function is Interval it follows that $J$ is an interval.
By the Corollary to Limit of Increasing Function and the Corollary to Limit of Decreasing Function, $f^{-1}: J \to I$ is continuous.
Next its derivative is to be considered.
Suppose $f$ is strictly increasing.
Let $y \in J^o$.
Then $\map {f^{-1} } y \in I^o$.
Let $k = \map {f^{-1} } {y + h} - \map {f^{-1} } y$.
Thus:
:$\map {f^{-1} } {y + h} = \map {f^{-1} } y + k = x + k$
Thus:
:$y + h = \map f {x + k}$
and hence:
:$h = \map f {x + k} - y = \map f {x + k} - \map f x$
Since $f^{-1}$ is continuous on $J$, it follows that $k \to 0$ as $h \to 0$.
Also, $f^{-1}$ is strictly increasing from Inverse of Strictly Monotone Function‎ and so $k \ne 0$ unless $h = 0$.
So by Limit of Composite Function we get:
:$\dfrac {\map {f^{-1} } {y + h} - \map {f^{-1} } y} h = \dfrac k {\map f {x + k} - \map f x}$
Thus:
:$\dfrac {\map {f^{-1} } {y + h} - \map {f^{-1} } y} h \to \dfrac 1 {D \map f x}$
as $h \to 0$.
Suppose $f$ is strictly decreasing.
Exactly the same argument applies.
== Also presented as ==
== Sources ==
* : $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Leibniz's Theorem for Differentiation of a Product: $3.3.9$
* : $\S 13$: General Rules of Differentiation: $13.12$
* : $5$. Differential Calculus: Appendix: Differentiation Rules: $6.$ Inverse functions
* : $\S 10.15 \ (5)$
* : $\S 12.10$
* 
Category:Derivative of Inverse Function
Category:Differential Calculus
Derivative of Inverse Function/Also presented as

== Derivative of Inverse Function: Also presented as ==
When Leibniz's notation for derivatives $\paren {\dfrac {\d y} {\d x} }$ is being used, Derivative of Inverse Function is usually seen presented as:
:$\dfrac {\d x} {\d y} = \dfrac 1 {\frac {\d y} {\d x} }$
or:
:$\dfrac {\d x} {\d y} = \dfrac 1 {\d y / \d x}$
where:
:$\dfrac {\d x} {\d y}$ is the derivative of $x$ $y$
:$\dfrac {\d y} {\d x}$ is the derivative of $y$ $x$.
This must not be interpreted to mean that derivative can be treated as fractions; it is simply a convenient notation.
== Sources ==
* : $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Leibniz's Theorem for Differentiation of a Product: $3.3.9$
* : $\S 13$: General Rules of Differentiation: $13.12$
* : $5$. Differential Calculus: Appendix: Differentiation Rules: $6.$ Inverse functions
* : $\S 12.10$
* 
Category:Derivative of Inverse Function
Derivative of Inverse Hyperbolic Cosecant

== Theorem ==
Let $x \in \R_{\ne 0}$.
Let $\arcsch x$ denote the inverse hyperbolic cosecant of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\arcsch x} = \dfrac {-1} {\size x \sqrt {1 + x^2} }$
== Proof ==
Let $x > 1$.
Then we have:
We have that Inverse Hyperbolic Cosecant is Odd Function.
Hence from Derivative of Odd Function is Even, $\map {\dfrac \d {\d x} } {\arcsch x}$ is even.
Hence for $x < -1$ we have that:
:$\map {\dfrac \d {\d x} } {\arcsch x} = \dfrac {-1} {\paren {-x} \sqrt {1 + x^2} }$
and so for $x < -1$:
:$\map {\dfrac \d {\d x} } {\arcsch x} = \dfrac {-1} {\size x \sqrt {1 + x^2} }$
and the result follows.
== Sources ==
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Cosecant
Derivative of Inverse Hyperbolic Cosecant Function

== Theorem ==
Let $u$ be a differentiable real function of $x$ such that $u \ne 0$.
Then:
:$\map {\dfrac \d {\d x} } {\arcsch u} = \dfrac {-1} {\size u \sqrt {1 + u^2} } \dfrac {\d u} {\d x}$ 
where $\arcsch$ is the real inverse hyperbolic cosecant.
== Proof ==
== Also see ==
* Derivative of Inverse Hyperbolic Sine Function
* Derivative of Real Area Hyperbolic Cosine of Function
* Derivative of Inverse Hyperbolic Tangent Function
* Derivative of Inverse Hyperbolic Cotangent Function
* Derivative of Inverse Hyperbolic Secant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.42$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions: $42$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $30$
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Cosecant
Derivative of Inverse Hyperbolic Cotangent

== Theorem ==
Let $S$ denote the union of the unbounded open real intervals:
:$S := \openint \gets {-1} \cup \openint 1 \to$
Let $x \in S$.
Let $\coth^{-1} x$ be the inverse hyperbolic cotangent of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac {-1} {x^2 - 1}$
== Proof ==
== Also presented as ==
This result can also be (and usually is) reported as:
:$\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac 1 {1 - x^2}$
but this obscures the fact that $x^2 > 1$ in order for it to be defined.
== Sources ==
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $7.$ Inverse hyperbolic trigonometric functions
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
Category:Derivative of Inverse Hyperbolic Cotangent
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Cotangent
Derivative of Inverse Hyperbolic Cotangent Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$ 
where $\size u > 1$
where $\coth^{-1}$ is the inverse hyperbolic cotangent.
== Proof ==
== Also presented as ==
Can also be seen (and in fact most often seen) presented in the form:
:$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$ 
but this obscures the fact that $\size u > 1$.
The condition is also often seen presented as $u^2 > 1$
== Also see ==
* Derivative of Inverse Hyperbolic Sine Function
* Derivative of Real Area Hyperbolic Cosine of Function
* Derivative of Inverse Hyperbolic Tangent Function
* Derivative of Inverse Hyperbolic Secant Function
* Derivative of Inverse Hyperbolic Cosecant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.40$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions: $40$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $28$
Category:Derivative of Inverse Hyperbolic Cotangent
Derivative of Inverse Hyperbolic Secant

== Theorem ==
Let $S$ denote the open real interval:
:$S := \openint 0 1$
Let $x \in S$.
Let $\arsech x$ denote the real inverse hyperbolic secant of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\arsech x} = \dfrac {-1} {x \sqrt{1 - x^2} }$
== Proof ==
$\arsech x$ is defined only on the half-open real interval $\hointl 0 1$.
Thus on $\hointl 0 1$:
When $x = 1$, however, $\sqrt{1 - x^2} = 0$ and so $\dfrac {-1} {x \sqrt {1 - x^2} }$ is undefined.
Hence $\arsech x$ can be defined only on $\openint 0 1$.
== Sources ==
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Secant
Derivative of Inverse Hyperbolic Secant Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\sech^{-1} u} = \dfrac {-1} {u \sqrt {1 - u^2} } \dfrac {\d u} {\d x}$ 
where $0 < u < 1$
where $\sech^{-1}$ is the inverse hyperbolic secant.
== Proof ==
== Also see ==
* Derivative of Inverse Hyperbolic Sine Function
* Derivative of Real Area Hyperbolic Cosine of Function
* Derivative of Inverse Hyperbolic Tangent Function
* Derivative of Inverse Hyperbolic Cotangent Function
* Derivative of Inverse Hyperbolic Cosecant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.41$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions: $41$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $29$
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Secant
Derivative of Inverse Hyperbolic Sine

== Theorem ==
Let $x \in \R$ be a real number.
Let $\arsinh x$ denote the inverse hyperbolic sine of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\arsinh x} = \dfrac 1 {\sqrt {x^2 + 1} }$
== Proof ==
For all $x \in \R$ we have that $\cosh y \ge 1$.
Thus it follows that it is necessary to take the positive root of $\sqrt {\sinh^2 y + 1}$.
So:
== Sources ==
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $7.$ Inverse hyperbolic trigonometric functions
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* 
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Sine
Category:Derivative of Inverse Hyperbolic Sine
Derivative of Inverse Hyperbolic Sine Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\arsinh u} = \dfrac 1 {\sqrt {1 + u^2} } \dfrac {\d u} {\d x}$ 
where $\sinh^{-1}$ is the inverse hyperbolic sine.
== Proof ==
== Also see ==
* Derivative of Real Area Hyperbolic Cosine of Function
* Derivative of Inverse Hyperbolic Tangent Function
* Derivative of Inverse Hyperbolic Cotangent Function
* Derivative of Inverse Hyperbolic Secant Function
* Derivative of Inverse Hyperbolic Cosecant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.37$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions: $37$.
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $25$
Category:Derivative of Inverse Hyperbolic Sine
Derivative of Inverse Hyperbolic Tangent

== Theorem ==
Let $S$ denote the open real interval:
:$S := \openint {-1} 1$
Let $x \in S$.
Let $\tanh^{-1} x$ be the inverse hyperbolic tangent of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\tanh^{-1} x} = \dfrac 1 {1 - x^2}$
== Proof ==
== Sources ==
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $7.$ Inverse hyperbolic trigonometric functions
* 
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* 
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivatives of Inverse Hyperbolic Functions
Category:Inverse Hyperbolic Tangent
Category:Derivative of Inverse Hyperbolic Tangent
Derivative of Inverse Hyperbolic Tangent Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\artanh u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$ 
where $\size u < 1$
where $\tanh^{-1}$ is the real area hyperbolic tangent.
== Proof ==
== Also see ==
* Derivative of Inverse Hyperbolic Sine Function
* Derivative of Real Area Hyperbolic Cosine of Function
* Derivative of Inverse Hyperbolic Cotangent Function
* Derivative of Inverse Hyperbolic Secant Function
* Derivative of Inverse Hyperbolic Cosecant Function
== Sources ==
* : $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.39$
* : Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions: $39$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $27$
Category:Derivative of Inverse Hyperbolic Tangent
Derivative of Laplace Transform

== Theorem ==
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any closed interval $\closedint 0 a$.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then, everywhere that $\dfrac \d {\d s} \laptrans f$ exists:
:$\dfrac \d {\d s} \laptrans {\map f t} = -\laptrans {t \, \map f t}$
== Proof ==
== Examples ==
== Also see ==
* Laplace Transform of Derivative
* Higher Order Derivatives of Laplace Transform
== Sources ==
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.10$
* : $\S 6.1$
Category:Laplace Transforms
Derivative of Laplace Transform/Examples/Example 1

== Example of Use of Derivative of Laplace Transform ==
:$\ds \int_0^\infty t e^{-2 t} \cos t \rd t = \dfrac 3 {25}$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Evaluation of Integrals: $45 \ \text{(a)}$
Category:Examples of Laplace Transforms
Derivative of Logarithm at One

== Theorem ==
Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
Category:Derivative of Logarithm at One
Category:Derivatives involving Logarithm Functions
Category:Logarithms
Derivative of Logarithm at One/Proof 1

== Theorem ==
== Proof ==
L'Hôpital's Rule gives:
:$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \lim_{x \mathop \to c} \frac {\map {f'} x} {\map {g'} x}$
(provided the appropriate conditions are fulfilled).
Here we have:
Thus:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = \lim_{x \mathop \to 0} \frac {\paren {1 + x}^{-1} } 1 = \frac 1 {1 + 0} = 1$
== Sources ==
* : $\S 14.5 \ (3) \ \text{(ii)}$
Category:Derivative of Logarithm at One
Derivative of Matrix Exponential

== Theorem ==
Let $\mathbf A$ be a square matrix.
Let $t \in \R$ be a real number.
Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.
Then:
:$\dfrac \d {\d t} e^{\mathbf A t} = \mathbf A e^{\mathbf A t}$
== Proof ==
From the definition of the matrix exponential, $e^{\mathbf A t}$ is defined as being the square matrix $X$ with the properties:
:$(1): \quad \map {\dfrac \d {\d t} } X = \mathbf A X$
:$(2): \quad \map X {\mathbf 0_n} = \mathbf I_n$
The result follows directly;
Category:Matrix Exponential
Category:Derivatives
Derivative of Monotone Function

== Theorem ==
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
=== Real Function with Positive Derivative is Increasing ===
=== Real Function with Strictly Positive Derivative is Strictly Increasing ===
=== Real Function with Negative Derivative is Decreasing ===
=== Real Function with Strictly Negative Derivative is Strictly Decreasing ===
== Sources ==
* : $\S 12.7$
Category:Differential Calculus
Derivative of Natural Logarithm Function

== Theorem ==
Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Examples ==
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $2$.
* : $\text {II}$. Calculus: Differentiation
* : Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $5.$ Logarithmic function
* : Appendix $2$: Table of derivatives and integrals of common functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
* 
Category:Derivative of Natural Logarithm Function
Category:Derivatives involving Logarithm Functions
Category:Natural Logarithms
Derivative of Natural Logarithm Function/Examples/(x-2)^1/3 times (x-3)^1/2 times (2x-1)^3/2

== Examples of Use of Derivative of Natural Logarithm Function ==
Let $y = \paren {x - 2}^{1/3} \paren {x - 3}^{1/2} \paren {2 x - 1}^{3/2}$.
Then:
:$\dfrac {\d y} {\d x} = \paren {\dfrac 1 {3 \paren {x - 2} } + \dfrac 1 {2 \paren {x - 3} } + \dfrac 3 {2 x - 1} } \paren {x - 2}^{1/3} \paren {x - 3}^{1/2} \paren {2 x - 1}^{3/2}$
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Differentiation: Variable Index: Example
Category:Derivative of Natural Logarithm Function