Question

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

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

Short answer

The required answer is$\frac{25}{6}{x}^{\frac{19}{6}}$

Step-by-step solution

Step 1. Given Information   

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

Step 2. Calculation    

Differentiate both the sides with respect to x, we get,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}\left(x^{1+2+\frac{1}{2}+\frac{2}{3}}\right) \\ =\frac{d}{dx}\left(x^{\frac{6+12+3+4}{6}}\right) \\ =\frac{d}{dx}{x}^{\frac{25}{6}} \\ =\frac{25}{6}{x}^{\frac{25}{6}-1} \\ =\frac{25}{6}{x}^{\frac{19}{6}} \end{gathered}$$