Question

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.

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

Short answer

The derivative of function is$y\text{'}=\frac{2x^{\ln x}\ln x}{x}$

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=x^{\ln x}$

Step 2. Calculation    

First, we will take the log on both sides and then differentiate it,

$$\begin{gathered} \ln y=\ln x^{\ln x} \\ \ln y=\ln x\ln x \\ \frac{1}{y}y\text{'}=\frac{\ln x}{x}+\frac{\ln x}{x} \\ \frac{1}{y}y\text{'}=\frac{2\ln x}{x} \\ y\text{'}=\frac{2x^{\ln x}\ln x}{x} \end{gathered}$$