Discussion
The derivative of x loge(x) is
Answer: D
The derivative is \(1 + \log_{e}(x)\).
Use the product rule with \(u = x\) and \(v = \log_{e}x\):
\[ \begin{aligned} \frac{d}{dx}(x\log_{e}x) &= 1\cdot\log_{e}x + x\cdot\frac{1}{x} \\ &= \log_{e}x + 1 \end{aligned} \]
The answer 1 keeps only the \(x\cdot\frac{1}{x}\) term and forgets to differentiate \(x\). The answers 0 and e are constants, but the derivative changes with \(x\): at \(x = 1\) it is 1 and at \(x = e\) it is 2.
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