Which of the following statements are true?

I. $\frac{d}{dx}\left(5^{x}\right) = x \cdot 5^{x - 1}$.

II. $\frac{d}{dx}\left(\log_{10} x\right) = \frac{1}{x\log 10}$.

III. $\frac{d}{dx}\left(e^{-3x}\right) = -3e^{-3x}$.

IV. $\frac{d}{dx}\left(\log 7x\right) = \frac{7}{x}$.

Explanation

1. 01

   II: $\log_{10} x = \frac{\log x}{\log 10}$, whose derivative is $\frac{1}{x\log 10}$. III: the chain rule gives the factor $-3$.

2. 02

   I is false: the power rule is for a variable base; $\frac{d}{dx}5^{x} = 5^{x}\log 5$. IV is false: $\log 7x = \log 7 + \log x$, so its derivative is $\frac{1}{x}$.