# Derivative of Hyperbolic Secant

## Theorem

$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$

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

## Proof 1

 $\ds \map {\frac \d {\d x} } {\sech x}$ $=$ $\ds \map {\frac \d {\d x} } {\frac 1 {\cosh x} }$ Definition of Hyperbolic Secant $\ds$ $=$ $\ds \map {\frac \d {\d x} } {\paren {\cosh x}^{-1} }$ Exponent Laws $\ds$ $=$ $\ds -\paren {\cosh x}^{-2} \sinh x$ Derivative of Hyperbolic Cosine, Power Rule for Derivatives, Chain Rule for Derivatives $\ds$ $=$ $\ds \frac {-1} {\cosh x} \frac {\sinh x} {\cosh x}$ Exponent Combination Laws $\ds$ $=$ $\ds -\sech z \tanh z$ Definition of Hyperbolic Secant and Definition of Hyperbolic Tangent

$\blacksquare$

## Proof 2

 $\ds \map {\frac \d {\d x} } {\sech x}$ $=$ $\ds 2 \map {\frac \d {\d x} } {\frac {e^x} {e^{2 x} + 1} }$ Definition of Hyperbolic Secant $\ds$ $=$ $\ds \frac 2 {\paren {e^{2 x} + 1}^2} \paren {\map {\frac \d {\d x} } {e^x} \paren {e^{2 x} + 1} - e^x \map {\frac \d {\d x} } {e^{2 x} + 1} }$ Quotient Rule for Derivatives $\ds$ $=$ $\ds -\frac 2 {\paren {e^{2 x} + 1}^2} \paren {2 e^{2 x} \cdot e^x - e^x \cdot e^{2 x} - e^x}$ Derivative of Exponential Function $\ds$ $=$ $\ds -\frac {2 \paren {e^{3 x} - e^x} } {\paren {e^{2 x} + 1}^2}$ $\ds$ $=$ $\ds -\frac {2 e^x} {\paren {e^{2 x} + 1} } \cdot \frac {e^{2 x} - 1} {e^{2 x} + 1}$ $\ds$ $=$ $\ds -\sech x \tanh x$ Definition of Hyperbolic Secant, Definition of Hyperbolic Tangent

$\blacksquare$