Question

Find the derivative using composition of functions of

`\sqrt[3]{ {x}^(2) }`

Answer 1

`\bf \sqrt[3]{x^2}\implies \left(\stackrel{outer~function}{\left( \stackrel{inner~function}{x} \right) }\right)^{(2)/(3)}\implies \stackrel{\qquad \qquad chain~rule}{\stackrel{outer}{\cfrac{2}{3}x^{-(1)/(3)}}}\cdot \stackrel{inner}{1} \\\\\\ \cfrac{2}{3x^{(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^{(2)/(3)} \end{cases}\implies g(~~f(x)~~)=(x)^{(2)/(3)}`