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.
Proof
| \(\ds \map {\frac \d {\d x} } {\coth^{-1} u}\) | \(=\) | \(\ds \map {\frac \d {\d u} } {\coth^{-1} u} \frac {\d u} {\d x}\) | Chain Rule for Derivatives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {-1} {u^2 - 1} \frac {\d u} {\d x}\) | Derivative of Inverse Hyperbolic Cotangent |
$\blacksquare$
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
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Hyperbolic and Inverse Hyperbolic Functions: $13.40$
- 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: Definitions of Inverse Hyperbolic Functions: $40$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $28$