Question

(d) If \( y = x^x \), then find \( \frac{dy}{dx} \).

Hint:

For expressions like \( y = x^x \), use logarithmic differentiation.

Answer 1

Take the logarithm of both sides:

\[ \ln y = x \ln x. \]

Differentiate with respect to \( x \):

\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1. \]

Multiply through by \( y = x^x \):

\[ \frac{dy}{dx} = x^x (\ln x + 1). \]