Derivative of Hyperbolic Cosine

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
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