Question

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

$f\left(x\right)=\mathrm{In}\left(x^x\right)$

Short answer

The required answer is$\ln x+1$.

Step-by-step solution

Step 1. Given information.

The given function is:

$f\left(x\right)=\mathrm{In}\left(x^x\right)$

Step 2. Find the derivative of the given function.

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