Derivative of Inverse Hyperbolic Cotangent Function

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

Then:
 * $\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$

where $\size u > 1$ where $\coth^{-1}$ is the inverse hyperbolic cotangent.

Also presented as
Can also be seen (and in fact most often seen) presented in the form:


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

but this obscures the fact that $\size u > 1$.

The condition is also often seen presented as $u^2 > 1$

Also see

 * Derivative of Inverse Hyperbolic Sine Function
 * Derivative of Real Area Hyperbolic Cosine of Function


 * Derivative of Inverse Hyperbolic Tangent Function


 * Derivative of Inverse Hyperbolic Secant Function
 * Derivative of Inverse Hyperbolic Cosecant Function