Question

Find \( \frac{dy}{dx} \), if \( y = (\log x)^x \).

Hint:

For differentiation of functions in the form \( y = f(x)^{g(x)} \), take the logarithm first: \[ \ln y = g(x) \ln f(x). \]

Answer 1

Step 1: Take the Logarithm
Taking the natural logarithm on both sides: \[ \ln y = x \ln (\log x). \] Step 2: Differentiate Both Sides
Differentiating using implicit differentiation: \[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} (x \ln (\log x)). \] Using the product rule: \[ \frac{d}{dx} (x \ln (\log x)) = x \times \frac{1}{\log x} \times \frac{1}{x} + \ln (\log x). \] \[ = \frac{1}{\log x} + \ln (\log x). \] Step 3: Solve for \( \frac{dy}{dx} \)
\[ \frac{dy}{dx} = y \left( \frac{1}{\log x} + \ln (\log x) \right). \] Substituting \( y = (\log x)^x \): \[ \frac{dy}{dx} = (\log x)^x \left( \frac{1}{\log x} + \ln (\log x) \right). \]