\begin{align*}
f(x)
&= {e}^{x} \cdot \sin{\left(x\right)}\\[0.75em]
f^\prime(x)
&= {\left[{e}^{x}\right]}^{\prime} \cdot \sin{\left(x\right)} + {e}^{x} \cdot {\left[\sin{\left(x\right)}\right]}^{\prime} \\[0.75em]
&= {e}^{x} \cdot \sin{\left(x\right)} + {e}^{x} \cdot \cos{\left(x\right)} \\[0.75em]
&= {e}^{x} \cdot \left( \sin{\left(x\right)} + \cos{\left(x\right)} \right)
\end{align*}