Derivative of Hyperbolic Cosine
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Theorem
- $\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$
where $\cosh$ is the hyperbolic cosine and $\sinh$ is the hyperbolic sine.
Proof
\(\ds \map {\dfrac \d {\d x} } {\cosh x}\) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {\dfrac {e^x + e ^{-x} } 2}\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \map {\dfrac \d {\d x} } {e^x + e^{-x} }\) | Derivative of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {e^x + \paren {-e^{-x} } }\) | Derivative of Exponential Function, Chain Rule for Derivatives, Linear Combination of Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {e^x - e^{-x} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sinh x\) | Definition of Hyperbolic Sine |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $9$.
- 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): cosh or ch
- 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): hyperbolic function
- 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 Cosine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosine.html