Q. 51 - Calculus

Question

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.     

$f\left(x\right)=\frac{x^2}{\sqrt[5]{x^2}}$

Step-by-Step Solution

Answer

The derivative of the function is $f\text{'}\left(x\right)=\frac{8}{5}{x}^{\frac{3}{5}}$.

Step 1. Given Information

The given function is $f\left(x\right)=\frac{x^2}{\sqrt[5]{x^2}}$.

Step 2. Simplify the function

Simplify the function.

$\begin{array}{rcl}f\left(x\right)& =& \frac{x^2}{x^{{\displaystyle \frac{2}{5}}}}\\ & =& x^{2-\frac{2}{5}}\\ & =& x^{\frac{8}{5}}\end{array}$

Step 3. Find the derivative

Apply the power rule of derivative, $\left(x^n\right)\text{'}=n{x}^{n-1}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{8}{5}{x}^{\frac{8}{5}-1}\\ & =& \frac{8}{5}{x}^{\frac{3}{5}}\end{array}$