Single Variable Calculus

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

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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} $

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