Derivative of Hyperbolic Sine

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
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