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

$\blacksquare$


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