Derivatives

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