Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 52

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

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