Derivative of Inverse Hyperbolic Tangent
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Theorem
Let $S$ denote the open real interval:
- $S := \openint {-1} 1$
Let $x \in S$.
Let $\tanh^{-1} x$ be the inverse hyperbolic tangent of $x$.
Then:
- $\map {\dfrac \d {\d x} } {\tanh^{-1} x} = \dfrac 1 {1 - x^2}$
Proof
\(\ds y\) | \(=\) | \(\ds \tanh^{-1} x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \tanh y\) | Definition of Real Inverse Hyperbolic Tangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d y}\) | \(=\) | \(\ds \sech^2 y\) | Derivative of Hyperbolic Tangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac 1 {\sech^2 y}\) | Derivative of Inverse Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac 1 {1 - \tanh^2 y}\) | Sum of Squares of Hyperbolic Secant and Tangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {\frac \d {\d x} } {\tanh^{-1} x}\) | \(=\) | \(\ds \frac 1 {1 - x^2}\) | Definition of $x$ and $y$ |
$\blacksquare$
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): arc-tanh
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse hyperbolic function
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives