Find the derivative of $f( x) =4e^{x}+x^{3}$.

Solution The function $f$ is the sum of $4e^{x}$ and $x^{3}$. Then $$\begin{eqnarray*} f^\prime (x) &=&\dfrac{d}{dx}(4e^{x}+x^{3}) {=} \dfrac{d}{dx}(4e^{x})+\dfrac{d}{dx}x^{3}{=}4\dfrac{d}{dx}e^{x}+3x^{2}{=}4e^{x}+3x^{2}\\[-6.4pt] &&\hspace{3.25pc}\underset{\color{#0066A7}{\scriptsize \hbox{Sum Rule}}}{\color{#0066A7}{\uparrow }}\hspace{1.9pc}\underset{\underset{\color{#0066A7}{\scriptsize \hbox{Simple Power Rule}}}{\color{#0066A7}{\scriptsize \hbox{Constant Multiple Rule}}}}{\color{#0066A7}{\uparrow }}\hspace{1.0pc}\underset{\color{#0066A7}{\scriptsize \hbox{Use (1)}.}}{\color{#0066A7}{\uparrow }} \end{eqnarray*}$$