Question

Find the derivatives of each of functions in Exercises 17–44. In some cases it may be convenient to do some preliminary algebra.

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

Short answer

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

Step-by-step solution

Step 1. Given information.

The given function is:

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

Step 2. Find the derivative of the given function.

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