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.

$f\left(x\right)={\left(x^{\frac{1}{3}}-2x\right)}^{-1}$

Short answer

The required answer is$-{\left(x^{\frac{1}{3}}-2x\right)}^{-2}\left(\frac{1}{3}{x}^{-\frac{2}{3}}-2\right)$

Step-by-step solution

Step 1. Given Information  

The given function is$f\left(x\right)={\left(x^{\frac{1}{3}}-2x\right)}^{-1}$

Step 2. Calculation    

Differentiate both the sides with respect tox, we get,

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