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.

$f\left(x\right)={\left(x\sqrt{x+1}\right)}^{-2}$

Short answer

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

Step-by-step solution

Step 1. Given Information 

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

Step 2. Calculation   

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

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