Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 6

Differentiate the function $f(x) = \log_5 \left( xe^x\right)$

$
\begin{equation}
\begin{aligned}
f'(x) &= \frac{d}{dx} \log_5 \left( xe^x \right)\\
\\
f'(x) &= \frac{1}{xe^x \ln 5} \cdot \frac{d}{dx} \left( x e^x \right)\\
\\
f'(x) &= \frac{1}{xe^x \ln 5} \left[ x \frac{d}{dx} \left( e^x \right) + e^x \frac{d}{dx} (x) \right]\\
\\
f'(x) &= \frac{1}{xe^x \ln 5} \left( xe^x + e^x \right)\\
\\
f'(x) &= \frac{\cancel{e^x}(x+1)}{x\cancel{e^x}\ln 5}\\
\\
f'(x) &= \frac{x+1}{x \ln 5}
\end{aligned}
\end{equation}
$