Question

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

Hint:

Logarithmic differentiation simplifies equations involving powers of variables.

Answer 1

Take the natural logarithm of both sides: \[ \ln (\cos x)^y = \ln (\cos y)^x. \] Differentiating both sides: \[ y \ln (\cos x) + \frac{y}{\cos x} \frac{dy}{dx} = x \ln (\cos y) + \frac{x}{\cos y}. \] Simplify to find \( \frac{dy}{dx} \).