Single Variable Calculus, Chapter 7, 7.4-2, Section 7.4-2, Problem 44

Suppose that $f(x) = x^{\cos x}$, find $f'(x)$. Illustrate the graphs of $f$ and $f'$.


$
\begin{equation}
\begin{aligned}

y =& x^{\cos x}
\\
\\
\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{1}{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}
$