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.

$f\left(x\right)=3{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{2}{3}}$

Short answer

The required answer is$-2{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{5}{3}}\left(16x{\left(x^2+1\right)}^7-7\right)$

Step-by-step solution

Step 1. Given Information    

The given function is$f\left(x\right)=3{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{2}{3}}$

Step 2. Calculation     

Differentiate both the sides with respect tox, we get,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}3{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{2}{3}} \\ =3\left(-\frac{2}{3}\right){\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{2}{3}-1}\frac{d}{dx}\left({\left(x^2+1\right)}^8-7x\right) \\ =-2{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{5}{3}}\left(8{\left(x^2+1\right)}^{8-1}\left(2x\right)-7\right) \\ =-2{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{5}{3}}\left(8{\left(x^2+1\right)}^7\left(2x\right)-7\right) \\ =-2{\left({\left(x^2+1\right)}^8-7x\right)}^{-\frac{5}{3}}\left(16x{\left(x^2+1\right)}^7-7\right) \end{gathered}$$