Question

1. $y = (ln x)^{ln x}$

Explanation:

Step1: Take natural logarithm on both sides

$$\ln y=\ln((\ln x)^{\ln x})=\ln x\cdot\ln(\ln x)$$

Step2: Differentiate both sides with respect to \(x\)

Using the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = \ln x\) and \(v=\ln(\ln x)\).

First, \(u^\prime=\frac{1}{x}\), and for \(v=\ln(\ln x)\), using the chain rule \(v^\prime=\frac{1}{\ln x}\cdot\frac{1}{x}\)

So \(\frac{y^\prime}{y}=\frac{1}{x}\cdot\ln(\ln x)+\ln x\cdot\frac{1}{\ln x}\cdot\frac{1}{x}\)

Step3: Simplify the right - hand side

\(\frac{y^\prime}{y}=\frac{\ln(\ln x)}{x}+\frac{1}{x}=\frac{\ln(\ln x) + 1}{x}\)

Step4: Solve for \(y^\prime\)

Multiply both sides by \(y = (\ln x)^{\ln x}\)

\(y^\prime=(\ln x)^{\ln x}\cdot\frac{\ln(\ln x)+1}{x}\)

Answer:

\(y^\prime = (\ln x)^{\ln x}\cdot\frac{\ln(\ln x)+1}{x}\)