Derivation of hyperbolic functions

Derivative calculator

Input field for the function:

f(x) =

\frac{d}{d x} f (x)

\frac{d^{n}}{d x^{n}} f (x)

Plot

Pos1

End

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e^{a x}

sin

cos

tan

e^x

x^a

\frac{a}{x}

asin

acos

atan

x²

√x

a^x

\frac{a}{x + b}

|x|

sinh

cosh

tanh

\coth

arsinh

arcosh

artanh

arcoth

a \cdot \sinh (b x + c)

a \cdot \cosh (b x + c)

a \cdot \tanh (b x + c)

\frac{1}{\sinh (x)}

a \cdot \cosh^{2} (b x + c)

a \cdot \tanh^{2} (b x + c)

\frac{1}{\cosh (x)}

\frac{\sinh (a \cdot x)}{\cosh (b \cdot x)}

e^{x} \sinh (x) \cosh (x)

\sinh (\cosh (x))

\frac{1}{\tanh (x)}

a \cdot \sinh^{2} (b x + c)

Function	Description
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 a^b
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:

$\frac{d}{d x} f (x) = \frac{d f}{d x} (x) = \frac{d f (x)}{d x} = f^{'} (x)$

Derivatives of hyperbolic functions and area functions

$\frac{d}{d x} \sinh (x) = \cosh (x)$

$\frac{d}{d x} \cosh (x) = \sinh (x)$

$\frac{d}{d x} \tanh (x) = \frac{1}{\cosh^{2} (x)}$

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

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

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

$\frac{d}{d x} artanh (x) = \frac{1}{1 - x^{2}}$

$\frac{d}{d x} arcoth (x) = - \frac{1}{1 - x^{2}}$

Derivation rules in short

Factor rule: A constant factor is preserved when differentiate

$(a \cdot f)' = a \cdot f'$

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

$(f_{1} + f_{2})' = f_{1}' + f_{2}'$

Product rule: Rule for deriving products

$(u \cdot v)' = u' \cdot v + u \cdot v'$

Quotient rule: Rule for deriving quotients

${(\frac{u}{v})}^{'} = \frac{u^{'} \cdot v - u \cdot v^{'}}{v^{2}}$

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

$(f (g (x)))' = f' (g) \cdot g' (x)$

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