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.

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

Short answer

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

Step-by-step solution

Step 1. Given Information   

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

Step 2. Calculation   

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

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