Derivative of Hyperbolic Cotangent
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Theorem
- $\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x = \dfrac {-1} {\sinh^2 x}$
where $\coth x$ denotes the hyperbolic cotangent, $\csch x$ denotes the hyperbolic cosecant and $\sinh x$ denotes the hyperbolic sine.
Corollary
- $\map {\dfrac \d {\d x} } {\coth x} = 1 - \coth^2 x$
Proof
\(\ds \map {\dfrac \d {\d x} } {\coth x}\) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {\frac {\cosh x} {\sinh x} }\) | Definition 2 of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh x \dfrac \d {\d x} \cosh x - \cosh x \dfrac \d {\d x} \sinh x} {\sinh^2 x}\) | Quotient Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh x \sinh x - \cosh x \dfrac \d {\d x} \cosh x} {\sinh^2 x}\) | Derivative of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh x \sinh x - \cosh x \cosh x} {\sinh^2 x}\) | Derivative of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {\sinh^2 x}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds -\csch^2 x\) | Definition 2 of Hyperbolic Cosecant |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $11$.
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $6.$ Hyperbolic trigonometric functions
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives
- Weisstein, Eric W. "Hyperbolic Cotangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCotangent.html