Differentiating an Expression Involving \(y=e^{x}\)

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

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