Q. 60 - 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.         

$f\left(x\right)=x^3\sqrt{x}\left(x^{\frac{2}{3}}\right)$

Step-by-Step Solution

Answer

The derivative of the function is $\frac{25}{6}{x}^{\frac{19}{6}}$.

Step 1. Given Information

The given function is $f\left(x\right)=x^3\sqrt{x}\left(x^{\frac{2}{3}}\right)$.

Step 1. Simplify the function

Simplify the given function.

$\begin{array}{rcl}f\left(x\right)& =& x^3{x}^{\frac{1}{2}}{x}^{\frac{2}{3}}\\ & =& x^{3+\frac{1}{2}+\frac{2}{3}}\\ & =& x^{\frac{18+3+4}{6}}\\ & =& x^{\frac{25}{6}}\end{array}$

Step 3. Find the derivative

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

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{25}{6}{x}^{\frac{25}{6}-1}\\ & =& \frac{25}{6}{x}^{\frac{19}{6}}\end{array}$