Discussion
If \(y = \log_{10}x + \log_{e}x\) , then \(\frac{dy}{dx}\) is equal to
Answer: E
\(\frac{dy}{dx} = \frac{1}{x}\left[\frac{1}{\log_{e}10} + 1\right]\).
Change the base: \(\log_{10}x = \frac{\log_{e}x}{\log_{e}10}\). Then
\[ \begin{aligned} \frac{dy}{dx} &= \frac{1}{x\log_{e}10} + \frac{1}{x} \\ &= \frac{1}{x}\left[\frac{1}{\log_{e}10} + 1\right] \end{aligned} \]
Since \(\frac{1}{\log_{e}10} = \log_{10}e\), this also equals \(\frac{1 + \log_{10}e}{x}\), which is not among the options: option B has \(\log_{e}10\) where \(\log_{10}e\) is needed, and option A has the wrong sign. The options without \(\frac{1}{x}\) are not derivatives of logarithms at all.
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