derivatives of hyperbolic functions

In this entry we compute the derivative of the hyperbolic functions $\sinh (x)$ and $\cosh (x)$.

Recall that by definition:

$\sinh (x)$$:=$${\displaystyle \frac{e^x-e^{-x}}{2}}$

$\cosh (x)$$:=$${\displaystyle \frac{e^x+e^{-x}}{2}}.$

Therefore:

${\displaystyle \frac{d}{dx}}\sinh (x)$$=$${\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{e^x-e^{-x}}{2}}\right)$

$=$${\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{d}{dx}}\left(e^x-e^{-x}\right)$

$=$${\displaystyle \frac{1}{2}}\cdot \left(e^x-(-e^{-x})\right)$

$=$${\displaystyle \frac{e^x+e^{-x}}{2}}$

$=$$\cosh (x).$

Similarly ${\displaystyle \frac{d}{dx}}\cosh (x)=\sinh (x)$. Using the quotient rule, we compute the derivative of $\tanh (x)={\displaystyle \frac{\sinh (x)}{\cosh (x)}}$:

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

where we have used the equality ${\cosh }^2(x)-{\sinh }^2(x)=1$.