Derivative of Inverse Hyperbolic Cotangent
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Theorem
Let $S$ denote the union of the unbounded open real intervals:
- $S := \openint \gets {-1} \cup \openint 1 \to$
Let $x \in S$.
Let $\coth^{-1} x$ be the inverse hyperbolic cotangent of $x$.
Then:
- $\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac {-1} {x^2 - 1}$
Proof
\(\ds y\) | \(=\) | \(\ds \coth^{-1} x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \coth y\) | Definition of Real Inverse Hyperbolic Cotangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d y}\) | \(=\) | \(\ds -\csch^2 y\) | Derivative of Hyperbolic Cotangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac {-1} {\csch^2 y}\) | Derivative of Inverse Function | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {\coth^2 y - 1}\) | Difference of Squares of Hyperbolic Cotangent and Cosecant | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {\frac \d {\d x} } {\coth^{-1} x}\) | \(=\) | \(\ds \frac {-1} {x^2 - 1}\) | Definition of $x$ and $y$ |
$\blacksquare$
Also presented as
This result can also be (and usually is) reported as:
- $\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac 1 {1 - x^2}$
but this obscures the fact that $x^2 > 1$ in order for it to be defined.
Sources
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $7.$ Inverse hyperbolic trigonometric functions
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives