Question

Find the derivative using composition of functions of 2%7D%20%7D%20" id="TexFormula1" title=" \sqrt[3]{ {x}^{2} } " alt=" \sqrt[3]{ {x}^{2} } " align="absmiddle" class="latex-formula">

Answer 1

$\bf \sqrt[3]{x^2}\implies \left(\stackrel{outer~function}{\left( \stackrel{inner~function}{x} \right) }\right)^{\frac{2}{3}}\implies \stackrel{\qquad \qquad chain~rule}{\stackrel{outer}{\cfrac{2}{3}x^{-\frac{1}{3}}}}\cdot \stackrel{inner}{1} \\\\\\ \cfrac{2}{3x^{\frac{1}{3}}}\implies \cfrac{2}{3\sqrt[3]{x}}$

so, the function is really a composite function, you could think of it 

$\bf \begin{cases} f(x)=x\\ g(x)=x^{\frac{2}{3}} \end{cases}\implies g(~~f(x)~~)=(x)^{\frac{2}{3}}$