Derivative of Inverse Hyperbolic Sine
Jump to navigation
Jump to search
Theorem
Let $x \in \R$ be a real number.
Let $\arsinh x$ denote the inverse hyperbolic sine of $x$.
Then:
- $\map {\dfrac \d {\d x} } {\arsinh x} = \dfrac 1 {\sqrt {x^2 + 1} }$
Proof
\(\ds y\) | \(=\) | \(\ds \arsinh x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \sinh y\) | Definition of Real Inverse Hyperbolic Sine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d y}\) | \(=\) | \(\ds \cosh y\) | Derivative of Hyperbolic Sine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac 1 {\cosh y}\) | Derivative of Inverse Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \pm \frac 1 {\sqrt {\sinh^2 y + 1} }\) | Difference of Squares of Hyperbolic Cosine and Sine |
For all $x \in \R$ we have that $\cosh y \ge 1$.
Thus it follows that it is necessary to take the positive root of $\sqrt {\sinh^2 y + 1}$.
So:
\(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac 1 {\sqrt {\sinh^2 y + 1} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {\frac \d {\d x} } {\arsinh x}\) | \(=\) | \(\ds \frac 1 {\sqrt {x^2 + 1} }\) | 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): 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