Question

(b) Find the differential coefficient of the function \( x^x \) with respect to \( x \):

Hint:

For \( x^x \), take the natural logarithm to simplify differentiation.

Answer 1

Let \( y = x^x \). Taking the natural logarithm on both sides: \[ \ln y = x \ln x. \] Differentiating both sides with respect to \( x \): \[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1. \] Multiply by \( y \) to get \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = x^x (\ln x + 1). \]