# derivatives of hyperbolic functions

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

Recall that by definition:

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

Therefore:

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

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

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

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

Title derivatives of hyperbolic functions DerivativesOfHyperbolicFunctions 2013-03-22 14:32:18 2013-03-22 14:32:18 alozano (2414) alozano (2414) 5 alozano (2414) Derivation msc 26A09