# Derivative of Hyperbolic Secant Function

## Theorem

$D_z \left({\operatorname{sech} z}\right) = -\operatorname{sech} z \ \tanh z$

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

## Proof 1

 $\displaystyle D_z \left({\sech z}\right)$ $=$ $\displaystyle D_z \left({\frac 1 {\cosh z} }\right)$ Definition of Hyperbolic Secant $\displaystyle$ $=$ $\displaystyle D_z \left({\left({\cosh z}\right)^{-1} }\right)$ Exponent Laws $\displaystyle$ $=$ $\displaystyle -\left({\cosh z}\right)^{-2} \, \sinh z$ Derivative of Hyperbolic Cosine Function, Power Rule for Derivatives, Chain Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \frac {-1} {\cosh z} \, \frac {\sinh z} {\cosh z}$ Exponent Combination Laws $\displaystyle$ $=$ $\displaystyle -\sech z \, \tanh z$ Definition of Hyperbolic Secant and Definition of Hyperbolic Tangent

$\blacksquare$

## Proof 2

 $\displaystyle \mathrm D_z \left(\operatorname{sech} z \right)$ $=$ $\displaystyle 2\mathrm D_z \left(\frac {e^z} {e^{2z} + 1} \right)$ Definition of Hyperbolic Secant $\displaystyle$ $=$ $\displaystyle \frac 2 {\left( {e^{2z} + 1}\right)^2} \left(\mathrm D_z\left[e^{z}\right]\left[e^{2z} + 1\right] - e^{z}\mathrm D_z\left[e^{2z} + 1\right]\right)$ Quotient Rule for Derivatives $\displaystyle$ $=$ $\displaystyle - \frac 2 {\left( {e^{2z} + 1}\right)^2} \left(2e^{2z} \cdot e^z - e^{z}\cdot e^{2z} - e^z\right)$ Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle - \frac {2\left(e^{3z} - e^{z}\right)} {\left( {e^{2z} + 1} \right)^2}$ $\displaystyle$ $=$ $\displaystyle - \frac {2e^z} {\left( {e^{2z} + 1} \right)} \cdot \frac {e^{2z}-1} {e^{2z} + 1}$ $\displaystyle$ $=$ $\displaystyle -\operatorname{sech} z \tanh z$ Definition of Hyperbolic Secant, Definition of Hyperbolic Tangent

$\blacksquare$