Derivatives of Hyperbolic and Inverse Hyperbolic Functions

The hyperbolic functions for a real number $x$ are defined as follows: \[\sinh x=\frac{1}{2}(e^x-e^{-x})\] \[\cosh x=\frac{1}{2}(e^x+e^{-x})\] \[\tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}\]$\operatorname{cosech}x$, $\operatorname{sech}x$, $\coth x$ are the reciprocals of $\sinh x$, $\cosh x$ and $\tanh x$ respectively and are defined accordingly. In the cases of $\operatorname{cosech}x$ and $\coth x$, $x$ cannot be zero. The derivatives of hyperbolic and inverse hyperbolic functions are given below:

Derivatives of Hyperbolic Functions

Derivative of $\sinh x$

\[\text{Let }y=\sinh x=\frac{1}{2}(e^x-e^{-x})\] \[\therefore \frac{dy}{dx}=\frac{1}{2}\frac{d(e^x-e^{-x})}{dx}\] \[=\frac{1}{2}(e^x+e^{-x})=\cosh x\] \[\therefore \frac{d(\sinh x)}{dx}=\cosh x\]


Derivative of $\cosh x$

\[\text{Let }y=\cosh x=\frac{1}{2}(e^x+e^{-x})\] \[\therefore \frac{dy}{dx}=\frac{1}{2}\frac{d(e^x+e^{-x})}{dx}\] \[=\frac{1}{2}(e^x-e^{-x})=\sinh x\] \[\therefore \frac{d(\cosh x)}{dx}=\sinh x\]


Derivative of $\tanh x$

\[\text{Let }y=\tanh x=\frac{\sinh x}{\cosh x}\] \[\therefore \frac{dy}{dx}\] \[=\frac{\cosh x\frac{d(\sinh x)}{dx}-\sinh x\frac{d(\cosh x)}{dx}}{\cosh^2x}\] \[=\frac{\cosh^2x-\sinh^2x}{\cosh^2x}\] \[=\frac{1}{\cosh^2x}=\operatorname{sech}^2x\] \[\therefore \frac{d(\tanh x)}{dx}=\operatorname{sech}^2x\]


Derivative of $\coth x$

\[\text{Let }y=\coth x=\frac{\cosh x}{\sinh x}\] \[\therefore \frac{dy}{dx}\] \[=\frac{\sinh x\frac{d(\cosh x)}{dx}-\cosh x\frac{d(\sinh x)}{dx}}{\sinh^2x}\] \[=\frac{\sinh^2x-\cosh^2x}{\sinh^2x}\] \[=\frac{-1}{\sinh^2x}=-\operatorname{cosech}^2x\]


Derivative of $\operatorname{sech}x$

\[\text{Let }y=\operatorname{sech}x=\frac{1}{\cosh x}\]\[=\frac{\cosh^2x-\sinh^2x}{\cosh x}\]\[=\cosh x-\sinh x\tanh x\] \[\therefore \frac{dy}{dx}=\frac{d(\cosh x)}{dx}-\frac{d}{dx}(\sinh x\tanh x)\] \[=\sinh x-\sinh x\frac{d(\tanh x)}{dx}-\tanh x\frac{d(\sinh x)}{dx}\] \[=\sinh x-\sinh x\operatorname{sech}^2x-\tanh x\cosh x\] \[=\sinh x-\frac{\sinh x}{\cosh x}\operatorname{sech}x-\sinh x\] \[=-\operatorname{sech}x\tanh x\] \[\therefore \frac{d(\operatorname{sech}x)}{dx}=-\operatorname{sech}x\tanh x\]


Derivative of $\operatorname{cosech}x$

\[\text{Let }y=\operatorname{cosech}x=\frac{1}{\sinh x}\]\[=\frac{\cosh^2x-\sinh^2x}{\sinh x}\] \[=\cosh x\coth x-\sinh x\] \[\therefore \frac{dy}{dx}=\frac{d}{dx}(\cosh x\coth x)-\frac{d(\sinh x)}{dx}\] \[=\cosh x\frac{d(\coth x)}{dx}+\coth x\frac{d(\cosh x)}{dx}-\cosh x\] \[=\cosh x×(-\operatorname{cosech}^2x)+\coth x×\sinh x-\cosh x\] \[=\frac{-\cosh x}{\sinh x}×\operatorname{cosech}x+\cosh x-\cosh x\]\[=-\operatorname{cosech}x\coth x\]


Derivatives of Inverse Hyperbolic Functions

Derivative of $\sinh^{-1}x$

\[\text{Let }y=\sinh^{-1}x\] \[x=\sinh y\] \[\frac{dx}{dy}=\frac{d(\sinh y)}{dy}=\cosh y\] \[=\sqrt{1+\sinh^2y}\] \[=\sqrt{1+x^2}\] \[\therefore \frac{dy}{dx}=\frac{1}{\sqrt{1+x^2}}\] \[\therefore \frac{d(\sinh^{-1}x)}{dx}=\frac{1}{\sqrt{1+x^2}}\]


Derivative of $\cosh^{-1}x$

\[\text{Let }y=\cosh^{-1}x\] \[x=\cosh y\] \[\frac{dx}{dy}=\frac{d(\cosh y)}{dy}=\sinh y\] \[=\sqrt{\cosh^2y-1}\] \[=\sqrt{x^2-1}\] \[\therefore \frac{dy}{dx}=\frac{1}{\sqrt{x^2-1}} \text{ }(x>1)\] \[\therefore \frac{d(\cosh^{-1}x)}{dx}=\frac{1}{\sqrt{x^2-1}}\text{ }(x>1)\]


Derivative of $\tanh^{-1}x$

\[\text{Let }y=\tanh^{-1}x\] \[x=\tanh y=\frac{\sinh y}{\cosh y}\] \[\frac{dx}{dy}=\frac{d}{dy}\left(\frac{\sinh y}{\cosh y}\right)\] \[=\frac{\cosh y\frac{d}{dy}(\sinh y)-\sinh y\frac{d}{dy}(\cosh y)}{\cosh^2y}\] \[=\frac{\cosh^2y-\sinh^2y}{\cosh^2y}\] \[=1-\tanh^2y\] \[=1-x^2\] \[\therefore \frac{dy}{dx}=\frac{1}{1-x^2}\text{ }(x<1)\] \[\therefore \frac{d(\tanh^{-1}x)}{dx}=\frac{1}{1-x^2}\text{ }(x<1)\]


Derivative of $\coth^{-1}x$

\[\text{Let }y=\coth^{-1}x\] \[x=\coth y=\frac{\cosh y}{\sinh y}\] \[\frac{dx}{dy}=\frac{d}{dy}\left(\frac{\cosh y}{\sinh y}\right)\] \[=\frac{\sinh y\frac{d}{dy}(\cosh y)-\cosh y\frac{d}{dy}(\sinh y)}{\sinh^2y}\] \[=\frac{\sinh^2y-\cosh^2y}{\sinh^2y}\] \[=1-\coth^2y\] \[=1-x^2\] \[=-(x^2-1)\] \[\therefore \frac{dy}{dx}=-\frac{1}{x^2-1}\text{ }(x>1)\] \[\therefore \frac{d(\coth^{-1}x)}{dx}=-\frac{1}{x^2-1}(x>1)\]


Derivative of $\operatorname{sech}^{-1}x$

\[\text{Let }y=\operatorname{sech}^{-1}x\] \[x=\operatorname{sech}y=\frac{1}{\cosh y}\] \[=\frac{\cosh^2x-\sinh^2x}{\cosh y}\] \[=\cosh y-\sinh y\tanh y\] \[\frac{dx}{dy}=\frac{d}{dy}(\cosh y-\sinh y\tanh y)\] \[=\sinh y-\sinh y\frac{d}{dy}(\tanh y)-\tanh y\frac{d}{dy}(\sinh y)\] \[=\sinh y-\sinh y\operatorname{sech}^2y-\tanh y\cosh y\] \[=\sinh y-\operatorname{sech}^2y\sqrt{\cosh^2y-1}-\sinh y\] \[=-\operatorname{sech}^2y\sqrt{\frac{1}{\operatorname{sech}^2y}-1}\] \[=-x^2\sqrt{\frac{1}{x^2}-1}\] \[=-x^2\sqrt{\frac{1-x^2}{x^2}}\] \[=-x\sqrt{1-x^2}\] \[\therefore \frac{dy}{dx}=-\frac{1}{x\sqrt{1-x^2}}\text{ }(x<1)\]


Derivative of $\operatorname{cosech}^{-1}x$

\[\text{Let }y=\operatorname{cosech}^{-1}x\]\[x=\operatorname{cosech y}=\frac{1}{\sinh y}\] \[=\frac{\cosh^2y-\sinh^2y}{\sinh y}\] \[=\cosh y\coth y-\sinh y\] \[\frac{dx}{dy}=\frac{d}{dy}(\cosh y\coth y-\sinh y)\] \[=\cosh y\frac{d}{dy}(\coth y)+\coth y\frac{d}{dy}(\cosh y)-\cosh y\] \[=-\cosh y\operatorname{cosech}^2y+\coth y\sinh y-\cosh y\] \[=-\operatorname{cosech}^2y\sqrt{1+\sinh^2y}+\cosh y-\cosh y\] \[=-\operatorname{cosech}^2y\sqrt{1+\frac{1}{\operatorname{cosech}^2y}}\] \[=-x^2\sqrt{1+\frac{1}{x^2}}\] \[=-x^2\sqrt{\frac{x^2+1}{x^2}}\] \[=-x\sqrt{x^2+1}\] \[\therefore \frac{dy}{dx}=-\frac{1}{x\sqrt{x^2+1}}\] \[\therefore \frac{d(\operatorname{cosech}^{-1}y)}{dx}=-\frac{1}{x\sqrt{x^2-1}}\]


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