Question

Differentiate \( y = (\cos x)^{\sin x \).}

Hint:

For \( y = f(x)^{g(x)} \), use logarithmic differentiation: \( \ln y = g(x) \ln f(x) \).

Answer 1

Step 1: Take natural logarithm on both sides. \[ \ln y = \sin x \ln (\cos x) \] Step 2: Differentiate both sides using chain rule. \[ \frac{1}{y} \frac{dy}{dx} = \cos x \ln (\cos x) + \frac{\sin x (-\sin x)}{\cos x} \] Step 3: Solve for \( \frac{dy}{dx} \). \[ \frac{dy}{dx} = y \left( \cos x \ln (\cos x) - \sin^2 x \right) \] Substituting \( y = (\cos x)^{\sin x} \), \[ \frac{dy}{dx} = (\cos x)^{\sin x} \left( \cos x \ln (\cos x) - \sin^2 x \right). \]