# Derivative of Hyperbolic Cosecant Function

## Theorem

$D_z \left({\operatorname{csch} z}\right) = -\operatorname{csch} z \ \coth z$

where $\coth z$ denotes the hyperbolic cotangent and $\operatorname{csch} z$ denotes the hyperbolic cosecant.

## Proof

 $\displaystyle D_z \left({\operatorname{csch} z}\right)$ $=$ $\displaystyle D_z \left({\frac 1 {\sinh z} }\right)$ $\quad$ Definition of Hyperbolic Cosecant $\quad$ $\displaystyle$ $=$ $\displaystyle D_z \left({\left({\sinh z}\right)^{-1} }\right)$ $\quad$ Exponent Laws $\quad$ $\displaystyle$ $=$ $\displaystyle -\left({\sinh z}\right)^{-2} \ \cosh z$ $\quad$ Derivative of Hyperbolic Cosine Function, Power Rule for Derivatives, Chain Rule $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {-1} {\sinh z} \ \frac {\cosh z} {\sinh z}$ $\quad$ Exponent Combination Laws $\quad$ $\displaystyle$ $=$ $\displaystyle -\operatorname{csch} z \ \coth z$ $\quad$ Definitions of Hyperbolic Cosecant and Hyperbolic Cotangent $\quad$

$\blacksquare$