# Derivation of the e-function

## Derivative Calculator

Function of x

First derivative of the function after x

Input field for the e-function:

f(x) =

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$\frac{d}{dx}f\left(x\right)$
$\frac{{d}^{n}}{d{x}^{n}}f\left(x\right)$
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${e}^{ax}$
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${e}^{\sqrt{x}}$
$a{e}^{-b{x}^{2}+c}$
$\sqrt{{e}^{ax}}$
$a{e}^{bx+c}$
${e}^{a{x}^{2}}$
$\frac{1}{{e}^{ax}}$
$\frac{x}{{e}^{x}}$
FunctionDescription
sin(x)Sine of x
cos(x)Cosine of x
tan(x)Tangent of x
asin(x)arcsine
acos(x)arccosine of x
atan(x)arctangent of x
atan2(y, x)Returns the arctangent of the quotient of its arguments.
cosh(x)Hyperbolic cosine of x
sinh(x)Hyperbolic sine of x
pow(a, b)Power ab
sqrt(x)Square root of x
exp(x)e-function
log(x), ln(x)Natural logarithm
log(x, b)Logarithm to base b
log2(x), lb(x)Logarithm to base 2
log10(x), ld(x)Logarithm to base 10
more ...

## Notations

Notations for derivatives:

$\frac{d}{dx}f\left(x\right)=\frac{df}{dx}\left(x\right)=\frac{df\left(x\right)}{dx}={f}^{\prime }\left(x\right)$

Notations of the natural exponential function:

${e}^{x}=\mathrm{exp}\left(x\right)$

The basis is the Euler number:

$e=2.71828182845904523536028747135266249...$

## Derivatives of the e-function

The e-function is the solution of the differential equation

$y\prime =y$

The chain rule for the e-function

$\frac{d}{dx}{e}^{f\left(x\right)}=f\prime \left(x\right)\cdot {e}^{f\left(x\right)}$

Derivations of the e-function

$\frac{d}{dx}{e}^{x}=\left({e}^{x}\right)\prime ={e}^{x}$

$\frac{d}{dx}{e}^{ax}$$=\left({e}^{ax}\right)\prime$$=a{e}^{ax}$

$\frac{d}{dx}{e}^{a{x}^{2}}$$=\left({e}^{a{x}^{2}}\right)\prime$$=2ax{e}^{a{x}^{2}}$

$\frac{d}{dx}\frac{1}{{e}^{x}}$$=\left(\frac{1}{{e}^{x}}\right)\prime$$=\left({e}^{-x}\right)\prime$$=-{e}^{-x}$$=-\frac{1}{{e}^{x}}$

$\frac{d}{dx}{e}^{\mathrm{ln}\left(x\right)}$$=\left({e}^{\mathrm{ln}\left(x\right)}\right)\prime$$=\left(x\right)\prime$$=1$

$\frac{d}{dx}{e}^{{x}^{n}}$$=\left({e}^{{x}^{n}}\right)\prime$$=n{x}^{n-1}{e}^{{x}^{n}}$

$\frac{d}{dx}{\left({e}^{x}\right)}^{n}$$=\left({\left({e}^{x}\right)}^{n}\right)\prime$$=\left({e}^{nx}\right)\prime$$=n{e}^{nx}$

Derivation of the logarithm functions

$\frac{d}{dx}\mathrm{ln}\left(x\right)=\frac{1}{x}$

$\frac{d}{dx}{\mathrm{log}}_{a}\left(x\right)=\frac{1}{x}{\mathrm{log}}_{a}\left(e\right)$

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