Question

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

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

Short answer

The derivative of function is$y\text{'}={\left(\ln x\right)}^{\ln x}\left[\frac{1}{x}+\frac{\ln \left(\ln x\right)}{x}\right]$

Step-by-step solution

Step 1. Given Information

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

Step 2. Calculation    

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

$$\begin{gathered} \ln y=\ln {\left(\ln x\right)}^{\ln x} \\ \ln y=\ln x·\ln \left(\ln x\right) \\ \frac{1}{y}y\text{'}=\ln x\frac{1}{\ln x}\frac{1}{x}+\ln \left(\ln x\right)\frac{1}{x} \\ \frac{1}{y}y\text{'}=\frac{1}{x}+\frac{\ln \left(\ln x\right)}{x} \\ y\text{'}={\left(\ln x\right)}^{\ln x}\left[\frac{1}{x}+\frac{\ln \left(\ln x\right)}{x}\right] \end{gathered}$$