# Derivative of Hyperbolic Sine Function

## Theorem

$\map {D_x} {\sinh x} = \cosh x$

where $\sinh$ is the hyperbolic sine and $\cosh$ is the hyperbolic cosine.

## Proof 1

 $\displaystyle \map {D_x} {\sinh x}$ $=$ $\displaystyle \map {D_x} {\dfrac {e^x - e ^{-x} } 2}$ Definition of Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \frac {\map {D_x} {e^x - e^{-x} } } 2$ Derivative of Constant Multiple $\displaystyle$ $=$ $\displaystyle \frac {\map {D_x} {e^x} - \map {D_x} {e^{-x} } } 2$ Linear Combination of Derivatives $\displaystyle$ $=$ $\displaystyle \frac {\map {D_x} {e^x} - \map {D_{-x} } {e^{-x} } \, \map {D_x} {-x} } 2$ Chain Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \frac {e^x - e^{-x} \map {D_x} {-x} } 2$ Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle \frac {e^x - e^{-x} \paren {-1} } 2$ Derivative of Identity Function $\displaystyle$ $=$ $\displaystyle \frac {e^x + e^{-x} } 2$ $\displaystyle$ $=$ $\displaystyle \cosh x$ Definition of Hyperbolic Cosine

$\blacksquare$

## Proof 2

 $\displaystyle D_x \left({\sinh x}\right)$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\sinh \left({x + h}\right) - \sinh x} h$ $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {2 \cosh \left({\frac {x + h + x} 2}\right) \sinh \left({\frac {x + h - x} 2}\right)} h$ Hyperbolic Sine minus Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {2 \cosh \left({x + \frac h 2}\right) \sinh \left({\frac h 2}\right)} h$ $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cosh \left({x + \frac h 2}\right) \sinh \left({\frac h 2}\right)} {\frac h 2}$ $\displaystyle$ $=$ $\displaystyle \lim_{2 d \mathop \to 0} \frac {\cosh \left({x + d}\right) \sinh \left({d}\right)} d$ where $d = \dfrac h 2$ $\displaystyle$ $=$ $\displaystyle \lim_{d \mathop \to 0} \frac {\cosh \left({x + d}\right) \sinh \left({d}\right)} d$ $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac {\sinh \left({d}\right)} d$ $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac { e^d - e^{-d} } {2 d}$ Definition of Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac { e^{2 d} - 1 } {2 d e^d}$ $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \frac { e^{2 d} - 1 } {2 d}$ $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \lim_{d \mathop \to 0} \frac { e^{2 d} - 1 } {2 d}$ $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \lim_{2 d \mathop \to 0} \frac { e^{2 d} - 1 } {2 d}$ $\displaystyle$ $=$ $\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d}$ Derivative of Exponential at Zero $\displaystyle$ $=$ $\displaystyle \cosh x$

$\blacksquare$

## Proof 3

 $\displaystyle \map {D_x} {\sinh x}$ $=$ $\displaystyle -i \, \map {D_x} {\sin i x}$ Hyperbolic Sine in terms of Sine $\displaystyle$ $=$ $\displaystyle \cos i x$ Derivative of Sine Function $\displaystyle$ $=$ $\displaystyle \cosh x$ Hyperbolic Cosine in terms of Cosine

$\blacksquare$