# Derivative of Hyperbolic Tangent Function

## Theorem

$\map {D_z} {\tanh z} = \sech^2 z$

where $\tanh$ is the hyperbolic tangent and $\sech$ is the hyperbolic secant.

## Proof

 $\displaystyle \map {D_z} {\tanh z}$ $=$ $\displaystyle \map {D_z} {\dfrac {\sinh z} {\cosh z} }$ Definition 2 of Hyperbolic Tangent $\displaystyle$ $=$ $\displaystyle \dfrac {\paren {D_z \sinh z} \cosh z - \sinh z \paren {D_z \cosh z} } {\cosh^2 z}$ Quotient Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \dfrac {\cosh^2 z - \sinh z \paren {D_z \cosh z} } {\cosh^2 z}$ Derivative of Hyperbolic Sine Function $\displaystyle$ $=$ $\displaystyle \dfrac {\cosh^2 z - \sinh^2 z} {\cosh^2 z}$ Derivative of Hyperbolic Cosine Function $\displaystyle$ $=$ $\displaystyle \dfrac 1 {\cosh^2 z}$ Difference of Squares of Hyperbolic Cosine and Sine $\displaystyle$ $=$ $\displaystyle \sech^2 z$ Definition 2 of Hyperbolic Secant

$\blacksquare$