Derivative of Hyperbolic Cosecant

Theorem

$\map {\dfrac \d {\d x} } {\csch x} = -\csch x \coth x$

where:

$\coth x$ denotes the hyperbolic cotangent and $\csch z$ denotes the hyperbolic cosecant.
$x \in \R_{\ne 0}$


Proof

It is noted that at $x = 0$, $\csch x$ is undefined.

Hence the restriction of the domain.

\(\ds \map {\dfrac \d {\d x} } {\csch x}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\frac 1 {\sinh x} }\) Definition of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \map {\dfrac \d {\d x} } {\paren {\sinh z}^{-1} }\) Exponent Laws
\(\ds \) \(=\) \(\ds -\paren {\sinh x}^{-2} \cosh x\) Derivative of Hyperbolic Cosine, Power Rule for Derivatives, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {-1} {\sinh x} \ \frac {\cosh x} {\sinh x}\) Exponent Combination Laws
\(\ds \) \(=\) \(\ds -\csch x \coth x\) Definition of Hyperbolic Cosecant and Definition of Hyperbolic Cotangent

$\blacksquare$


Also see


Sources

  • 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
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