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.

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

Short answer

The required answer is $\frac{x}{\sqrt{x^2+1}}$

Step-by-step solution

Step 1. Given Information  

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

Step 2. Calculation  

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

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