We need to find the derivative of \(\cosh(x)\):
\[\frac{d}{dx}\cosh(x) = \frac{d}{dx}\frac{e^x + e^{-x}}{2}\]
Since 2 is a constant:
\[\frac{d}{dx}\cosh(x) = \frac{1}{2} \left( \frac{d}{dx} e^x + e^{-x} \right) \]
We can derivative them individually:
\[\begin{align} \frac{d}{dx} \cosh(x) = \frac{1}{2} \left( \frac{d}{dx} e^x + \frac{d}{dx}e^{-x} \right)\\ \frac{d}{dx} \cosh(x) = \frac{1}{2} ( e^x - e^{-x} ) \end{align}\]
This is the definition of \(\sinh(x)\):
\[ \frac{d}{dx} \cosh(x) = \sinh(x)\]