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Derivatives of Inverse Hyperbolic functions

Inverse hyperbolic functions are six types and the differentiation rules of each inverse hyperbolic function with respect to $x$ is listed here along with its proof in calculus mathematics.


Inverse Hyperbolic Sine function

$\dfrac{d}{dx} \, \sinh^{-1} x = \dfrac{1}{\sqrt{1+x^2}}$


Inverse Hyperbolic Cosine function

$\dfrac{d}{dx} \, \cosh^{-1} x = \dfrac{1}{\sqrt{x^2 -1}}$


Inverse Hyperbolic Tangent function

$\dfrac{d}{dx} \, \tanh^{-1} x = \dfrac{1}{1-x^2}$

The derivative of inverse hyperbolic tangent function with respect to $x$ is equal to $1$ divided by $1$ minus $x$ squared.


Inverse Hyperbolic Cotangent function

$\dfrac{d}{dx} \, \coth^{-1} x = \dfrac{1}{1-x^2}$


Inverse Hyperbolic Secant function

$\dfrac{d}{dx} \, \operatorname{sech}^{-1} x = \dfrac{-1}{|x| \sqrt{1 -x^2}}$


Inverse Hyperbolic Cosecant function

$\dfrac{d}{dx} \, \operatorname{csch}^{-1} x = \dfrac{-1}{|x| \sqrt{x^2 +1}}$

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