Derivative of Hyperbolic Cosecant Function

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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

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

$\blacksquare$


Also see


Sources