Question

What's the derivative of cubed root X

Answer 1

$\large\begin{array}{l} \textsf{Finding the derivative of}\\\\ \mathsf{f(x)=\,^3\!\!\!\sqrt{x}}\\\\ \mathsf{f(x)=x^{\frac{1}{3}}}\\\\\\ \textsf{Differentiate it by applying the power rule:}\\\\ \mathsf{f'(x)=\dfrac{1}{3}\,x^{\frac{1}{3}-1}}\\\\ \mathsf{f'(x)=\dfrac{1}{3}\,x^{\frac{1}{3}-\frac{3}{3}}}\\\\ \mathsf{f'(x)=\dfrac{1}{3}\,x^{\frac{1-3}{3}}}\\\\ \mathsf{f'(x)=\dfrac{1}{3}\,x^{-\frac{2}{3}}} \end{array}$

$\large\begin{array}{l} \mathsf{f'(x)=\dfrac{1}{3}\cdot \dfrac{1}{x^{\frac{2}{3}}}}\\\\ \mathsf{f'(x)=\dfrac{1}{3x^{\frac{2}{3}}}}\\\\ \boxed{\begin{array}{c}\mathsf{f'(x)=\dfrac{1}{3\,^3\!\!\!\sqrt{x^2}}} \end{array}} \end{array}$


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Tags: derivative cube root function irrational power rule differential cauculus