Derivative of Hyperbolic Cosecant Function

Theorem

Let $u$ be a differentiable real function of $x$.

Then:

$\map {\dfrac \d {\d x} } {\csch u} = -\csch u \coth u \dfrac {\d u} {\d x}$

where $\coth x$ denotes the hyperbolic cotangent and $\csch x$ denotes the hyperbolic cosecant.


Proof

\(\ds \map {\frac \d {\d x} } {\csch u}\) \(=\) \(\ds \map {\frac \d {\d u} } {\csch u} \frac {\d u} {\d x}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds -\csch u \coth u \frac {\d u} {\d x}\) Derivative of Hyperbolic Cotangent

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.36$
  • 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Differentiation Formulas: $36$.
  • 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $24$
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