Derivative of Hyperbolic Cosecant
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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
- Weisstein, Eric W. "Hyperbolic Cosecant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosecant.html