Derivative of Hyperbolic Tangent

Theorem

$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x = \dfrac 1 {\cosh^2 x}$

where $\tanh$ is the hyperbolic tangent, $\sech$ is the hyperbolic secant and $\cosh$ is the hyperbolic cosine.


Corollary

$\map {\dfrac \d {\d x} } {\tanh x} = 1 - \tanh^2 x$


Proof

\(\ds \map {\dfrac \d {\d x} } {\tanh x}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\dfrac {\sinh x} {\cosh x} }\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \dfrac {\paren {\dfrac \d {\d x} \sinh x} \cosh x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}\) Quotient Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {\cosh^2 x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}\) Derivative of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \dfrac {\cosh^2 x - \sinh^2 x} {\cosh^2 x}\) Derivative of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \dfrac 1 {\cosh^2 x}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \sech^2 x\) Definition 2 of Hyperbolic Secant

$\blacksquare$


Also see


Sources

  • 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $10$.
  • 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): 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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