Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 46

Find the derivative of the function $y = x^{\cos x}$, using log differentiation

$\begin{equation} \begin{aligned} \ln y &= \ln x^{\cos x}\\ \\ \ln y &= \cos x \ln x\\ \\ \frac{d}{dx} \ln y &= \frac{d}{dx} (\cos x \ln x)\\ \\ \frac{1}{y} \frac{dy}{dx} &= \cos x \frac{d}{dx} (\ln x) + \ln x \frac{d}{dx} (\cos x)\\ \\ \frac{y'}{y} &= \cos x \cdot \frac{1}{x} + \ln x (- \sin x)\\ \\ y' &= y \left( \frac{\cos x}{x} - \sin x \ln x \right)\\ \\ y' &= x^{\cos x} \left(\frac{\cos x}{x} - \sin x \ln x \right) \end{aligned} \end{equation}$