\begin{equation*} 	\frac{d(\sinh x)}{dx} = \frac{d\left(\displaystyle\frac{e^{x} - e^{-x}}{2}\right)}{dx}	%\\ 		= \frac{1}{2}\cdot\frac{d\left(e^{x} - e^{-x}\right)}{dx}	%\\ 		= \frac{1}{2}\cdot\left(e^{x} + e^{-x}\right)	%\\ 		= \cosh x \end{equation*}