Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given Information

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

Step 2. Calculation    

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

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