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Wize University Calculus 1 Textbook > Derivatives

Inverse Hyperbolic Functions and their Derivatives

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Inverse Hyperbolic Functions and their Derivatives

We define the inverse hyperbolic functions is the usual way inverse functions are defined. For instance,

sinh⁡(x)=y  ⟺  x=arcsinh(y)\boxed{\displaystyle \sinh (x) = y \iff x= \arcsinh (y)}
This idea gives us the six inverse hyperbolic functions: arcsinhx, arcCoshx, arctanhx, arcCothx, arcsechx, and arcCschx.\arcsinh x, \ \arccosh x ,\ \arctanh x , \ \arccoth x, \ \arcsech x, \text{ and } \arccsch x.

Derivatives of Inverse Hyperbolic Functions

ddxarcsinhx=1x2+1ddxarcCoshx=1x2−1ddxarctanhx=11−x2\displaystyle \boxed{\frac{d}{dx}\arcsinh x = \frac{1}{\sqrt{x^2+1}} }\hspace{1cm} \boxed{ \frac{d}{dx}\arccosh x = \frac{1}{\sqrt{x^2-1}}} \hspace{1cm} \boxed{ \frac{d}{dx}\arctanh x = \frac{1}{1-x^2}}

          ddxarcCothx=11−x2ddxarcsechx=−1x1−x2ddxarcCschx=−1∣x∣1+x2\displaystyle \ \ \ \ \ \ \ \ \ \ \boxed{\frac{d}{dx}\arccoth x = \frac{1}{1-x^2}}\hspace{1 cm} \boxed{\frac{d}{dx}\arcsech x = -\frac{1}{x\sqrt{1-x^2}}} \hspace{1cm} \boxed{ \frac{d}{dx}\arccsch x = -\frac{1}{|x|\sqrt{1+x^2}}}

*NOTE: The video has incorrect signs on the formulas for arcCoth\arccoth and arcCsch\arccsch.
Find the derivative of f(x)=arcsech1−x2f(x) = \arcsech \sqrt{1-x^2}.