# Derivative of Hyperbolic Sine Function/Proof 1

$D_x \left({\sinh x}\right) = \cosh x$
 $\displaystyle D_x \left({ \sinh x }\right)$ $=$ $\displaystyle D_x \left({\dfrac {e^x - e ^{-x} } 2}\right)$ Definition of Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \frac {D_x \left({e^x - e^{-x} }\right)} 2$ Derivative of Constant Multiple $\displaystyle$ $=$ $\displaystyle \frac {D_x \left({e^x}\right) - D_x \left({ e^{-x} }\right)} 2$ Linear Combination of Derivatives $\displaystyle$ $=$ $\displaystyle \frac {D_x \left({e^x}\right) - D_{-x} \left({ e^{-x} }\right) D_x \left({-x}\right)} 2$ Chain Rule $\displaystyle$ $=$ $\displaystyle \frac {e^x - e^{-x} D_x \left({-x}\right)} 2$ Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle \frac {e^x - e^{-x} \left({-1}\right)} 2$ Derivative of Identity Function $\displaystyle$ $=$ $\displaystyle \frac {e^x + e^{-x} } 2$ $\displaystyle$ $=$ $\displaystyle \cosh x$ Definition of Hyperbolic Cosine
$\blacksquare$