Derivative of Hyperbolic Cotangent Function
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Theorem
Let $u$ be a differentiable real function of $x$.
Then:
- $\map {\dfrac \d {\d x} } {\coth u} = -\csch^2 u \dfrac {\d u} {\d x}$
where $\coth x$ denotes the hyperbolic cotangent and $\csch x$ denotes the hyperbolic cosecant.
Proof
\(\ds \map {\frac \d {\d x} } {\coth u}\) | \(=\) | \(\ds \map {\frac \d {\d u} } {\coth u} \frac {\d u} {\d x}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\csch^2 u \frac {\d u} {\d x}\) | 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: $13.34$
- 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: $34$.
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $22$