Derivative of Hyperbolic Sine
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Theorem
- $\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$
where $\sinh$ is the hyperbolic sine and $\cosh$ is the hyperbolic cosine.
Proof 1
\(\ds \map {\frac \d {\d x} } {\sinh x}\) | \(=\) | \(\ds \map {\frac \d {\d x} } {\dfrac {e^x - e ^{-x} } 2}\) | Definition of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\map {\frac \d {\d x} } {e^x} - \map {\frac \d {\d x} } {e^{-x} } }\) | Linear Combination of Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {e^x - \paren {-e^{-x} } }\) | Derivative of Exponential Function, Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^x + e^{-x} } 2\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x\) | Definition of Hyperbolic Cosine |
$\blacksquare$
Proof 2
\(\ds \map {\frac \d {\d x} } {\sinh x}\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \sinh {x + h} - \sinh x} h\) | Definition of Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {2 \map \cosh {\frac {x + h + x} 2} \map \sinh {\frac {x + h - x} 2} } h\) | Hyperbolic Sine minus Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {2 \map \cosh {x + \frac h 2} \map \sinh {\frac h 2} } h\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \cosh {x + \frac h 2} \map \sinh {\frac h 2} } {\frac h 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{2 d \mathop \to 0} \frac {\map \cosh {x + d} \map \sinh d} d\) | where $d = \dfrac h 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{d \mathop \to 0} \frac {\map \cosh {x + d} \map \sinh d} d\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac {\map \sinh d} d\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac {e^d - e^{-d} } {2 d}\) | Definition of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac {e^{2 d} - 1 } {2 d e^d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \frac {e^{2 d} - 1} {2 d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \lim_{d \mathop \to 0} \frac {e^{2 d} - 1} {2 d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \lim_{2 d \mathop \to 0} \frac {e^{2 d} - 1} {2 d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d}\) | Derivative of Exponential at Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x\) |
$\blacksquare$
Proof 3
\(\ds \map {\frac \d {\d x} } {\sinh x}\) | \(=\) | \(\ds -i \map {\frac \d {\d x} } {\sin i x}\) | Hyperbolic Sine in terms of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos i x\) | Derivative of Sine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x\) | Hyperbolic Cosine in terms of Cosine |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $8$.
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $6.$ 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: 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): 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
- Weisstein, Eric W. "Hyperbolic Sine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSine.html