Derivative of Real Area Hyperbolic Cosine
Jump to navigation
Jump to search
Theorem
Let $x \in \R_{>1}$ be a real number.
Let $\arcosh x$ be the real area hyperbolic cosine of $x$.
Then:
- $\map {\dfrac \d {\d x} } {\arcosh x} = \dfrac 1 {\sqrt {x^2 - 1} }$
Proof
\(\ds y\) | \(=\) | \(\ds \arcosh x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \cosh y\) | Definition of Real Area Hyperbolic Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d y}\) | \(=\) | \(\ds \sinh y\) | Derivative of Hyperbolic Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac 1 {\sinh y}\) | Derivative of Inverse Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \pm \frac 1 {\sqrt {\cosh^2 y - 1} }\) | Difference of Squares of Hyperbolic Cosine and Sine |
Note that when $y = 0$, $\cosh y$ is defined and equals $1$.
But from Derivative of Hyperbolic Cosine:
- $\valueat {\dfrac \d {\d y} \cosh y} {y \mathop = 0} = \sinh 0 = 0$
Thus $\dfrac {\d y} {\d x} = \dfrac 1 {\sinh y}$ is not defined at $y = 0$.
Hence the limitation of the domain of $\map {\dfrac \d {\d x} } {\arcosh x}$ to exclude $x = 1$.
Now it is necessary to determine the sign of $\dfrac {\d y} {\d x}$.
From:
- Real Area Hyperbolic Cosine is Strictly Increasing
- Derivative of Strictly Increasing Real Function is Strictly Positive
it follows that $\map {\dfrac \d {\d x} } {\arcosh x} > 0$ on $\R_{>1}$.
Thus:
- $\dfrac {\d y} {\d x} = \dfrac 1 {\sqrt {\cosh^2 y - 1} }$
where $\sqrt {\cosh^2 y - 1}$ denotes the positive square root of $\cosh^2 y - 1$.
Hence by definition of $x$ and $y$ above:
- $\map {\dfrac \d {\d x} } {\arcosh x} = \dfrac 1 {\sqrt {x^2 - 1} }$
$\blacksquare$
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