nyuuzyou/wikis huggingface dataset:
Derivative of Composite Function/Second Derivative

== Theorem ==
Let ${D_x}^k u$ denote the $k$th derivative of a function $u$ $x$.
Then:
:${D_x}^2 w = {D_u}^2 w \paren { {D_x}^1 u}^2 + {D_u}^1 w {D_x}^2 u$
== Proof ==
For ease of understanding, let Leibniz's notation be used:
:$\dfrac {\d^k u} {\d x^k} := {D_x}^k u$
Then we have:
== Sources ==
* : $\S 1.2.5$: Permutations and Factorials: Exercise $21$
Category:Derivative of Composite Function
Derivative of Composite Function/Third Derivative

== Theorem ==
Let ${D_x}^k u$ denote the $k$th derivative of a function $u$ $x$.
Then:
:${D_x}^3 w = {D_u}^3 w \paren { {D_x}^1 u}^3 + 3 {D_u}^2 w {D_x}^2 u {D_x}^1 u + {D_u}^1 w {D_x}^3 u$
== Proof ==
For ease of understanding, let Leibniz's notation be used:
:$\dfrac {\d^k u} {\d x^k} := {D_x}^k u$
Then we have:
Category:Derivative of Composite Function
Derivative of Concatenation of Complex Paths

== Theorem ==
Let $\gamma_0 : \closedint {a_0}{b_0} \to \C$ be an injective complex-differentiable function.
Let $\gamma_1 : \closedint {a_1}{b_1} \to \C$ be a complex-differentiable function such that $\map {\gamma_0}{b_0} = \map {\gamma_1}{a_1}$.
Let $t_0 \in \openint {a_0}{b_0}$.
Define the concatenation $\gamma_0 * \gamma_1 : \closedint 0 1 \to \C$ as:
:$\map {\gamma_0 * \gamma_1}{ s } = \begin{cases} \map {\gamma_0}{2 s \paren{b_0 - a_0} + a_0} & \forall s \in \closedint 0 {\dfrac 1 2} \\ \map {\gamma_1}{ \paren{2 s - 1} \paren{b_1 - a_1} + a_1} & \forall s \in \closedint {\dfrac 1 2} 1 \end{cases}$
Then there exists $s_0 \in \openint 0 {\dfrac 1 2}$ and $C \in \R_{>0}$ such that:
== Proof ==
Set $s_0 := \dfrac {t_0 - a_0}{2 \paren{ b_0 - a_0 } }$.
Then:
For all $s \in \openint 0 {\dfrac 1 2}$, it follows that:
Hence:
As $\closedint {a_0}{b_0}$ is a closed real interval, it follows that $a_0 < b_0$, so $2 \paren {b_0 - a_0} > 0$.
Category:Complex Differential Calculus
Derivative of Constant

== Theorem ==
Let $\map {f_c} x$ be the constant function on $\R$, where $c \in \R$.
Then:
:$\map { {f_c}'} x = 0$
=== Complex Domain ===
== Proof ==
The function $f_c: \R \to \R$ is defined as:
:$\forall x \in \R: \map {f_c} x = c$
Thus:
== Also see ==
This is the converse of Zero Derivative implies Constant Function.
Thus we see that $f$ is the constant function $\forall x: \map {f'} x = 0$.
== Sources ==
* : $\text {II}$. Calculus: Differentiation: Standard Differential Coefficients
* : $\S 13$: Derivatives: General Rules of Differentiation: $13.2$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $1$
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $1.$ Constant factor
* : $\S 15$: Derivatives: General Rules of Differentiation: $15.2.$
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivatives
Category:Constant Mappings
Derivative of Constant/Complex

== Theorem ==
Let $\map {f_c} z$ be the constant function on an open domain $D \in \C$, where $c \in \C$.
Then:
:$\forall z \in D : \map { {f_c}'} z = 0$
== Proof ==
The function $f_c: D \to \C$ is defined as:
:$\forall z \in D: \map {f_c} z = c$
Thus:
Category:Complex Differential Calculus
Category:Constant Mappings
Derivative of Constant Multiple/Complex

== Theorem ==
Let $D$ be an open subset of the set of complex numbers $\C$.
Let $f: D \to \C$ be a complex-differentiable function on $D$.
Let $c \in \C$ be a constant.
Then:
:$\forall z \in D : \map {D_z} {c \map f z} = c \map {D_z} {\map f z}$
== Proof ==
Category:Derivative of Constant Multiple
Category:Complex Analysis
Derivative of Constant Multiple/Real

== Theorem ==
Let $f$ be a real function which is differentiable on $\R$.
Let $c \in \R$ be a constant.
Then:
:$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$
=== Corollary ===
== Proof ==
== Sources ==
* : $\S 13$: General Rules of Differentiation: $13.6$
* : $5$. Differential Calculus: Appendix: Differentiation Rules: $1.$ Constant factor
Category:Derivative of Constant Multiple
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 ==
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 $\map {\dfrac {\d^k} {\d x^k} } {c \map f x} = c \map {\dfrac {\d^k} {\d x^k} } {\map f x}$:
Hence the result by induction.
Category:Derivative of Constant Multiple
Derivative of Constant to Power of Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Let $a \in \R_{>0}$.
Let $a^u$ be $a$ raised to the power of $u$.
Then:
:$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$
== Proof ==
== Sources ==
* : $\S 13$: Derivatives of Exponential and Logarithmic Functions: $13.28$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $11$
Category:Derivatives involving Exponential Function
Derivative of Cosecant Function

== Theorem ==
:$\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$
where $\sin x \ne 0$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Derivative of Sine Function
* Derivative of Cosine Function
* Derivative of Tangent Function
* Derivative of Cotangent Function
* Derivative of Secant Function
== Sources ==
* : $\text {II}$. Calculus: Differentiation
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivative of Cosecant Function
Category:Derivatives of Trigonometric Functions
Category:Cosecant Function
Derivative of Cosecant Function/Proof 1

== Theorem ==
== Proof ==
From the definition of the cosecant function:
:$\csc x = \dfrac 1 {\sin x}$
From Derivative of Sine Function:
:$\map {\dfrac \d {\d x} } {\sin x} = \cos x$
Then:
This is valid only when $\sin x \ne 0$.
Category:Derivative of Cosecant Function
Derivative of Cosecant of Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$
where $\sec$ is the cosecant function and $\tan$ is the cotangent function.
== Proof ==
== Also see ==
* Derivative of Sine of Function
* Derivative of Cosine of Function
* Derivative of Tangent of Function
* Derivative of Cotangent of Function
* Derivative of Secant of Function
== Sources ==
* : $\S 13$: Derivatives of Trigonometric and Inverse Trigonometric Functions: $13.19$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $8$
Category:Derivatives of Trigonometric Functions
Category:Cosecant Function
Derivative of Cosecant of a x

== Corollary to Derivative of Cosecant Function ==
:$\map {\dfrac \d {\d x} } {\csc a x} = -a \csc a x \cot a x$
== Proof ==
== Also see ==
* Derivative of $\sin a x$
* Derivative of $\cos a x$
* Derivative of $\tan a x$
* Derivative of $\cot a x$
* Derivative of $\sec a x$
Category:Derivatives of Trigonometric Functions
Category:Cosecant Function
Derivative of Cosine Function

== Theorem ==
:$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Also see ==
* Derivative of Sine Function
* Derivative of Tangent Function
* Derivative of Cotangent Function
* Derivative of Secant Function
* Derivative of Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $5$.
* : $\text {II}$. Calculus: Differentiation
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $3.$ Trigonometric functions
* 
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivatives of Trigonometric Functions
Category:Cosine Function
Category:Derivative of Cosine Function
Derivative of Cosine Function/Corollary

== Corollary to Derivative of Cosine Function ==
:$\map {\dfrac \d {\d x} } {\cos a x} = -a \sin a x$
== Proof ==
== Also see ==
* Derivative of $\sin a x$
* Derivative of $\tan a x$
* Derivative of $\cot a x$
* Derivative of $\sec a x$
* Derivative of $\csc a x$
Category:Derivative of Cosine Function
Derivative of Cosine Function/Proof 1

== Theorem ==
== Proof ==
From the definition of the cosine function, we have:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Then:
The result follows from the definition of the sine function.
== Sources ==
* : $\S 16.3 \ (1) \ \text{(v)}$
Category:Derivative of Cosine Function
Derivative of Cosine Integral Function

== Theorem ==
:$\dfrac \d {\d x} \paren {\map \Ci x} = -\dfrac {\cos x} x$
where:
:$\Ci$ denotes the cosine integral function
:$x$ is a strictly positive real number.
== Proof ==
== Also see ==
* Derivative of Sine Integral Function
Category:Cosine Integral Function
Category:Derivatives
Derivative of Cosine of Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\cos u} = -\sin u \dfrac {\d u} {\d x}$
where $\sin$ is the sine function and $\cos$ is the cosine function.
== Proof ==
== Also see ==
* Derivative of Sine of Function
* Derivative of Tangent of Function
* Derivative of Cotangent of Function
* Derivative of Secant of Function
* Derivative of Cosecant of Function
== Sources ==
* : $\S 13$: Derivatives of Trigonometric and Inverse Trigonometric Functions: $13.15$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $4$
Category:Derivative of Cosine Function
Derivative of Cotangent Function

== Theorem ==
:$\map {\dfrac \d {\d x} } {\cot x} = -\csc^2 x = \dfrac {-1} {\sin^2 x}$
where $\sin x \ne 0$.
=== Corollary 1 ===
=== Corollary 2 ===
=== Corollary 3 ===
== Proof ==
== Also see ==
* Derivative of Sine Function
* Derivative of Cosine Function
* Derivative of Tangent Function
* Derivative of Secant Function
* Derivative of Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $7$.
* : $\text {II}$. Calculus: Differentiation
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $3.$ Trigonometric functions
* : Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivative of Cotangent Function
Category:Derivatives of Trigonometric Functions
Category:Cotangent Function
Derivative of Cotangent Function/Corollary 1

== Corollary to Derivative of Cotangent Function ==
:$\map {\dfrac \d {\d x} } {\cot a x} = -a \csc^2 a x$
== Proof ==
== Also see ==
* Derivative of $\sin a x$
* Derivative of $\cos a x$
* Derivative of $\tan a x$
* Derivative of $\sec a x$
* Derivative of $\csc a x$
Category:Derivative of Cotangent Function
Derivative of Cotangent Function/Corollary 2

== Corollary to Derivative of Cotangent Function ==
:$\dfrac \d {\d x} \cot x = -1 - \cot^2 x$
== Proof ==
== Sources ==
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $3.$ Trigonometric functions
Category:Derivative of Cotangent Function
Derivative of Cotangent Function/Corollary 3

== Corollary to Derivative of Cotangent Function ==
:$\map {\dfrac \d {\d x} } {\cot a x} = -a \paren {\cot^2 a x + 1}$
== Proof ==
== Also see ==
* Derivative of $\sin a x$
* Derivative of $\cos a x$
* Derivative of $\tan a x$
* Derivative of $\sec a x$
* Derivative of $\csc a x$
Category:Derivative of Cotangent Function
Derivative of Cotangent Function/Proof

== Theorem ==
== Proof ==
From the definition of the cotangent function:
:$\cot x = \dfrac {\cos x} {\sin x}$
From Derivative of Sine Function:
:$\map {\dfrac \d {\d x} } {\sin x} = \cos x$
From Derivative of Cosine Function:
:$\map {\dfrac \d {\d x} } {\cos x}= -\sin x$
Then:
This is valid only when $\sin x \ne 0$.
The result follows from the definition of the real cosecant function.
== Sources ==
* : $\text {II}$. Calculus: Differentiation: Quotient
Category:Derivative of Cotangent Function
Derivative of Cotangent of Function

== Theorem ==
Let $u$ be a differentiable real function of $x$.
Then:
:$\map {\dfrac \d {\d x} } {\cot u} = -\csc^2 u \dfrac {\d u} {\d x}$
where $\cot$ is the cotangent function and $\csc$ is the cosecant function.
== Proof ==
== Also see ==
* Derivative of Sine of Function
* Derivative of Cosine of Function
* Derivative of Tangent of Function
* Derivative of Secant of Function
* Derivative of Cosecant of Function
== Sources ==
* : $\S 13$: Derivatives of Trigonometric and Inverse Trigonometric Functions: $13.17$
* : Chapter $4$. Derivatives: Derivatives of Special Functions: $6$
Category:Derivative of Cotangent Function
Derivative of Curve at Point

== Theorem ==
Let $f: \R \to \R$ be a real function.
Let the graph $G$ of $f$ be depicted on a Cartesian plane.
Then the derivative of $f$ at $x = \xi$ is equal to the tangent to $G$ at $x = \xi$.
== Proof ==
Let $f: \R \to \R$ be a real function.
:
Let the graph $G$ of $f$ be depicted on a Cartesian plane.
Let $A = \tuple {\xi, \map f \xi}$ be a point on $G$.
Consider the secant $AB$ to $G$ where $B = \tuple {\xi + h, \map f {\xi + h} }$.
From Slope of Secant, the slope of $AB$ is given by:
:$\dfrac {\map f {x + h} - \map f x} h$
By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$:
:$\map {f'} x = \ds \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h$
where $\lim$ denotes the limit as $h \to 0$.
== Sources ==
* : $\text {II}$. Calculus: Differentiation: Differential Coefficient
* : Chapter $8$: The System of the World: Calculus
Category:Analytic Geometry
Category:Differential Calculus
Derivative of Derivative is Subset of Derivative in T1 Space

== Theorem ==
Let $T = \struct {S, \tau}$ be a $T_1$ topological space.
Let $A$ be a subset of $S$.
Then
:$A' ' \subseteq A'$
where
:$A'$ denotes the derivative of $A$
== Proof ==
Let:
:$(1): \quad x \in A' '$
 $x \notin A'$.
Then by Characterization of Derivative by Open Sets there exists an open subset $G$ of $T$ such that:
:$(2): \quad x \in G$
and:
:$(3): \quad \lnot \exists y: y \in A \cap G \land x \ne y$
By definition of $T_1$ space:
:$\set x$ is closed.
Then by Open Set minus Closed Set is Open:
:$(4): \quad G \setminus \set x$ is open.
By $(1)$, $(2)$, and Characterization of Derivative by Open Sets there exists a point $y$ of $T$ such that:
:$(5): \quad y \in A' \cap G$
and:
:$(6): \quad x \ne y$
Then by definition of intersection:
:$y \in A'$
Then by $(6)$ and definition of set difference:
:$(7): \quad y \in A' \setminus \set x$
By definition of intersection and $(5)$:
:$y \in G$
By $(6)$ and definition of singleton:
:$y \notin \set x$
Then by definition of set difference:
:$(8): \quad y \in G \setminus \set x$
We will prove:
:$(9): \quad G \cap \paren {A \setminus \set x} = \O$
::
:: $G \cap \paren {A \setminus \set x} \ne \O$
:Then by definition of the empty set there exists $g$ such that:
::$g \in G \cap \paren {A \setminus \set x}$
:Hence by definition of intersection:
:: $g \in G$
:and:
:: $g \in A \setminus \set x$
:Then by definition of set difference:
::$g \in A$ 
:Hence by definition of intersection:
:: $g \in A \cap G$
:Then by $(3)$:
:: $x = g$
:This by definition of singleton contradicts with $g \notin \set x$ obtained by definition of set difference.
:Thus:
::$G \cap \paren {A \setminus \set x} = \O$
Define $U = G \setminus \set x$ as an open set by $(4)$.
By $(5)$ and definition of set difference:
:$y \in A'$
Then by $(8)$ and Characterization of Derivative by Open Sets there exists a point $q$ of $T$ such that:
:$(10): \quad q \in A \cap U$
and:
:$(11): \quad y \ne q$
By $(10)$ and definition of intersection:
:$q \in A$
By $(11)$ and definition of singleton:
:$q \notin \set y$
Then by definition of set difference
:$(12): \quad q \in A \setminus \set y$
By definition of intersection:
:$q \in U$
Then by $(12)$ and by definition of set difference
:$q \ne x$
and:
:$q \in A$
Then by definition of set difference:
:$q \in A \setminus \set x$
and :
:$q \in G$
Hence this contradicts with $(9)$ by definition of intersection.
Thus the result by Proof by Contradiction.
== Sources ==
* 
* 
Category:Set Derivatives
Category:T1 Spaces
Derivative of Differentiable Function on Open Interval is Baire Function

== Theorem ==
Let $I = \openint a b$ be a open interval.
Let $f : I \to \R$ be a differentiable function.
Let $f' : I \to \R$ be the derivative of $f$.
Then $f'$ is a Baire function. 
== Proof ==
For each $n \in \N$, let $f_n : I \to \R$ be a function with: 
:$\ds \map {f_n} x = n \paren {\map f {\min \set {x + \frac 1 n, \frac {b + x} 2} } - \map f x}$
for each $x$.
Note that from:
:Minimum of Finitely Many Continuous Real Functions is Continuous
:Composite of Continuous Mappings is Continuous
:Combined Sum Rule for Continuous Real Functions
we have:
:$f_n$ is continuous for each $n$. 
We aim to show that $f_n$ converges pointwise to $f'$.
Fix $x \in I$. 
Let $\epsilon$ be a positive real number.
Since $f$ is differentiable with derivative $f'$, there exists $\delta > 0$ such that: 
:$\ds \size {\frac {\map f {x + h} - \map f x} h - \map {f'} x} < \epsilon$
for all real $h$ with $0 < \size h < \delta$.
Let $N$ be the least natural number such that:
:$\ds x + \frac 1 N < \frac {b + x} 2$
That is:
:$\ds N > \frac 2 {b - x} > 0$
Then, for $n > N$ we have: 
:$\ds \min \set {x + \frac 1 n, \frac {b + x} 2} = x + \frac 1 n$
Then, for $n > \max \set {1/\delta, N}$ we also have $0 < 1/n < \delta$, and so: 
By the definition of pointwise convergence:
:$f_n$ converges pointwise to $f'$.
Since $\sequence {f_n}$ is a sequence of continuous real functions converging pointwise to $f'$, we have: 
:$f'$ is a Baire function.
Category:Baire Functions
Derivative of Differentiable Function on Open Interval is Somewhere Continuous

== Theorem ==
Let $I = \openint a b$ be a open interval..
Let $f : I \to \R$ be a differentiable function.
Let $f'$ be the derivative of $f$. 
Then $f'$ has a point of continuity.
== Proof ==
Let $D$ be the set of discontinuities of $f'$.
Let $d$ be Euclidean metric on $\R$. 
From Derivative of Differentiable Function on Open Interval is Baire Function:
:$f'$ is a Baire function.
From Set of Discontinuities of Baire Function is Meager:
:$D$ is meager in $\struct {I, d}$.
From Open Real Interval is Non-Meager:
:$I$ is non-meager in $\struct {I, d}$.
Therefore $D \ne I$. 
Since $D \subset I$, we have that $I \setminus D$ is non-empty.
That is, $f'$ has a point of continuity.
Category:Meager Spaces
Derivative of Dot Product of Vector-Valued Functions

== Theorem ==
Let $\mathbf a: \R \to \R^n$ and $\mathbf b: \R \to \R^n$ be differentiable vector-valued functions.
The derivative of their dot product is given by:
:$\map {\dfrac \d {\d x} } {\mathbf a \cdot \mathbf b} = \dfrac {\d \mathbf a} {\d x} \cdot \mathbf b + \mathbf a \cdot \dfrac {\d \mathbf b} {\d x}$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Derivative of Vector Cross Product of Vector-Valued Functions
* Derivative of Product of Real Function and Vector-Valued Function
== Sources ==
* : $\S 22$: Formulas involving Derivatives: $22.23$
Category:Product Rule for Derivatives
Category:Vector Calculus
Category:Dot Product
Category:Derivative of Dot Product of Vector-Valued Functions
Derivative of Dot Product of Vector-Valued Functions/Proof 1

== Theorem ==
== Proof ==
Let:
:$\mathbf a: x \mapsto \tuple {\map {a_1} x, \map {a_2} x, \ldots, \map {a_n} x}$
:$\mathbf b: x \mapsto \tuple {\map {b_1} x, \map {b_2} x, \ldots, \map {b_n} x}$
Then:
== Sources ==
* : Chapter $\text I$: Vector Algebra: $\S 7$
* : $\S 12.2$
Category:Derivative of Dot Product of Vector-Valued Functions
Derivative of Dot Product of Vector-Valued Functions/Proof 2

== Theorem ==
== Proof ==
Let $\mathbf v = \mathbf a \cdot \mathbf b$.
Then:
== Sources ==
* : Chapter $\text {III}$: The Differentiation of Vectors: $2$. Differentiation of Sums and Products: $(3.3)$
Category:Derivative of Dot Product of Vector-Valued Functions
Derivative of Error Function

== Theorem ==
:$\map {\dfrac \d {\d x} } {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$
where $\erf$ denotes the error function.
== Proof ==
We have, by the definition of the error function:
:$\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-t^2} \rd t$
By Fundamental Theorem of Calculus (First Part): Corollary, we therefore have:
:$\map {\dfrac \d {\d x} } {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$
Category:Error Function
Category:Derivatives
Derivative of Exponential Function

== Theorem ==
Let $\exp$ be the exponential function.
Then:
:$\map {\dfrac \d {\d x} } {\exp x} = \exp x$
=== Corollary 1 ===
=== Corollary 2 ===
=== Corollary 3 ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Proof 5 ==
== Also see ==
* Equivalence of Definitions of Exponential Function where it is shown that $\dfrac \d {\d x} \exp x = \exp x$ can be used to define the (real) exponential function.
== Sources ==
* : $\text {II}$. Calculus: Differentiation
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $5.$ Exponential function
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : Appendix $2$: Table of derivatives and integrals of common functions
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
* : $\S 5.4$
Category:Derivative of Exponential Function
Category:Derivatives involving Exponential Function
Category:Exponential Function
Derivative of Exponential Function/Complex

== Theorem ==
The complex exponential function is its own derivative.
That is:
:$\map {D_z} {\exp z} = \exp z$
== Proof from Sequence Definition ==
Take the definition of $\exp$ to be the limit of the sequence $\sequence {E_n}$ defined by:
:$\ds \map {E_n} z = \paren {1 + \dfrac z n}^n$
Then $\left \langle{E_n}\right \rangle$ is uniformly convergent on compact subsets of $\C$.
Further, $\C$ is an open, connected subset of $\C$.
So the hypotheses of Derivative of Sequence of Holomorphic Functions are satisfied.
Hence:
Hence the result.
Category:Derivatives involving Exponential Function
Derivative of Exponential Function/Corollary 1

== Corollary to Derivative of Exponential Function ==
Let $\exp x$ be the exponential function.
Let $a \in \R$.
Then:
:$\map {\dfrac \d {\d x} } {\map \exp {a x} } = a \map \exp {a x}$
== Proof ==
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $3$.
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Derivatives involving Exponential Function
Derivative of Exponential Function/Corollary 3

== Corollary to Derivative of Exponential Function ==
Let $\exp x$ be the exponential function.
:$\map {\dfrac \d {\d x} } {\map \exp {-x} } = -\map \exp {-x}$
== Proof ==
== Sources ==
* : Appendix: Table $1$: Derivatives
* : Appendix: Table $1$: Derivatives
Category:Derivatives involving Exponential Function
Derivative of Exponential Function/Proof 4

== Theorem ==
== Proof ==
This proof assumes the power series definition of $\exp$.
That is, let:
:$\ds \exp x = \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}$
From Series of Power over Factorial Converges, the interval of convergence of $\exp$ is the entirety of $\R$.
So we may apply Differentiation of Real Power Series to $\exp$ for all $x \in \R$.
Thus we have:
Hence the result.
Category:Derivative of Exponential Function