# 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

$\blacksquare$