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}
$