# Derivation of hyperbolic functions

## Derivative calculator

Input field for the 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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sin
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asin
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$\frac{a}{x+b}$
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sinh
cosh
tanh
$\mathrm{coth}$
$\mathrm{arsinh}$
$\mathrm{arcosh}$
$\mathrm{artanh}$
$\mathrm{arcoth}$
$a\cdot \mathrm{sinh}\left(bx+c\right)$
$a\cdot \mathrm{cosh}\left(bx+c\right)$
$a\cdot \mathrm{tanh}\left(bx+c\right)$
$\frac{1}{\mathrm{sinh}\left(x\right)}$
$a\cdot {\mathrm{cosh}}^{2}\left(bx+c\right)$
$a\cdot {\mathrm{tanh}}^{2}\left(bx+c\right)$
$\frac{1}{\mathrm{cosh}\left(x\right)}$
$\frac{\mathrm{sinh}\left(a\cdot x\right)}{\mathrm{cosh}\left(b\cdot x\right)}$
${e}^{x}\mathrm{sinh}\left(x\right)\mathrm{cosh}\left(x\right)$
$\mathrm{sinh}\left(\mathrm{cosh}\left(x\right)\right)$
$\frac{1}{\mathrm{tanh}\left(x\right)}$
$a\cdot {\mathrm{sinh}}^{2}\left(bx+c\right)$
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:

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

## Derivatives of hyperbolic functions and area functions

$\frac{d}{dx}\mathrm{sinh}\left(x\right)=\mathrm{cosh}\left(x\right)$

$\frac{d}{dx}\mathrm{cosh}\left(x\right)=\mathrm{sinh}\left(x\right)$

$\frac{d}{dx}\mathrm{tanh}\left(x\right)=\frac{1}{{\mathrm{cosh}}^{2}\left(x\right)}$

$\frac{d}{dx}\mathrm{coth}\left(x\right)=-\frac{1}{{\mathrm{sinh}}^{2}\left(x\right)}$

$\frac{d}{dx}\mathrm{arsinh}\left(x\right)=\frac{1}{\sqrt{1+{x}^{2}}}$

$\frac{d}{dx}\mathrm{arcosh}\left(x\right)=\frac{1}{\sqrt{{x}^{2}-1}}$

$\frac{d}{dx}\mathrm{artanh}\left(x\right)=\frac{1}{1-{x}^{2}}$

$\frac{d}{dx}\mathrm{arcoth}\left(x\right)=-\frac{1}{1-{x}^{2}}$

### Derivation rules in short

Factor rule: A constant factor is preserved when differentiate

$\left(a\cdot f\right)\prime =a\cdot f\prime$

Sum rule: When deriving a sum, the summands can be derived individually

$\left({f}_{1}+{f}_{2}\right)\prime ={f}_{1}\prime +{f}_{2}\prime$

Product rule: Rule for deriving products

$\left(u\cdot v\right)\prime =u\prime \cdot v+u\cdot v\prime$

Quotient rule: Rule for deriving quotients

${\left(\frac{u}{v}\right)}^{\prime }=\frac{{u}^{\prime }\cdot v-u\cdot {v}^{\prime }}{{v}^{2}}$

Chain rule: Nested functions go into a product of the inner and outer derivatives when differentiated

$\left(f\left(g\left(x\right)\right)\right)\prime =f\prime \left(g\right)\cdot g\prime \left(x\right)$

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