Calculus and Its Applications, Chapter 1, 1.5, Section 1.5, Problem 28

Determine the derivative $\displaystyle \frac{d}{dx} \left( \sqrt[3]{x} - \frac{4}{\sqrt{x}} \right)$

$
\begin{equation}
\begin{aligned}
\frac{d}{dx} \left( \sqrt[3]{x} - \frac{4}{\sqrt{x}} \right) &= \frac{d}{dx} \left( \sqrt[3]{x} \right) + \frac{d}{dx} \left( \frac{4}{\sqrt{x}} \right)\\
\\
&= \frac{d}{dx} \left( x^{\frac{1}{3}} \right) + \frac{d}{dx} \left( 4x^{-\frac{1}{2}} \right)\\
\\
&= \frac{1}{3} \cdot x^{\frac{1}{3} -1} + 4 \cdot \frac{d}{dx} \left( x^{-\frac{1}{2}} \right)\\
\\
&= \frac{1}{3} \cdot x^{\frac{-2}{3}} + 4 \cdot \frac{-1}{2} x^{\frac{-1}{2}}\\
\\
&= \frac{1}{3\sqrt[3]{x^2}} - 2x^{-\frac{3}{2}}\\
\\
&= \frac{1}{3\sqrt[3]{x^2}} - \frac{2}{x^{\frac{3}{2}}}\\
\\
&= \frac{1}{3\sqrt[3]{x^2}} - \frac{2}{\sqrt{x^3}}
\end{aligned}
\end{equation}
$