Question

Find derivative: ^6 square root x^5

Answer 1

$\large\begin{array}{l} \textsf{Find the derivative of}\\\\ \mathsf{f(x)=\,^6\!\!\!\sqrt{x^5}}\qquad\qquad\textsf{(expressing it as a power...)}\\\\ \mathsf{f(x)=x^{\frac{5}{6}}} \end{array}$


$\large\begin{array}{l} \textsf{So you can use the power rule to differentiate it:}\\\\ \mathsf{f'(x)=\big(x^{\frac{5}{6}}\big)'}\\\\ \mathsf{f'(x)=\dfrac{5}{6}\,x^{\frac{5}{6}-1}}\\\\ \mathsf{f'(x)=\dfrac{5}{6}\,x^{\frac{5}{6}-\frac{6}{6}}} \end{array}$

$\large\begin{array}{l} \mathsf{f'(x)=\dfrac{5}{6}\,x^{\frac{5-6}{6}}}\\\\ \mathsf{f'(x)=\dfrac{5}{6}\,x^{-\,\frac{1}{6}}}\\\\ \mathsf{f'(x)=\dfrac{5}{6}\cdot \dfrac{1}{x^{\frac{1}{6}}}}\\\\ \mathsf{f'(x)=\dfrac{5}{6x^{\frac{1}{6}}}}\\\\\\ \boxed{\begin{array}{c}\mathsf{f'(x)=\dfrac{5}{6\,^6\!\!\!\sqrt{x}}} \end{array}}\qquad\checkmark \end{array}$


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$\large\textsf{I hope it helps. :-)}$


Tags: derivative sixth root irrational function power rule differential calculus