Question

If \[ y = x^{\sin x}, \] find \(\frac{dy}{dx}\).

Hint:

Use logarithmic differentiation when the variable is both the base and the exponent.

Answer 1

Rewrite \( y \) using logarithms: \[ \ln y = \sin x \cdot \ln x. \] Differentiate both sides with respect to \( x \): \[ \frac{1}{y} \frac{dy}{dx} = \cos x \cdot \ln x + \sin x \cdot \frac{1}{x}. \] Multiply both sides by \( y \): \[ \frac{dy}{dx} = y \left( \cos x \ln x + \frac{\sin x}{x} \right). \] Substitute back \( y = x^{\sin x} \): \[ \boxed{ \frac{dy}{dx} = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right). } \]