Derivative of Hyperbolic Cotangent/Corollary

Theorem

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

where $\coth x$ denotes the hyperbolic cotangent.


Proof

\(\ds \dfrac \d {\d x} \coth x\) \(=\) \(\ds -\csch^2 x\) Derivative of Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds 1 - \coth^2 x\) Difference of Squares of Hyperbolic Cotangent and Cosecant

$\blacksquare$


Sources

  • 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
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