Question

Consider the following statements:
Statement 1: If $y=\log _{10} x+\log _{e} x$ then $\frac{d y}{d x}=\frac{\log _{10} e}{x} + \frac{1}{x}$
Statement $2: \frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}$ and $\frac{d}{d x}\left(\log _{e} x\right)=\frac{\log x}{\log e}$

• A. Statement 1 is true; statement 2 is false
• B. Statement 1 is false; statement 2 is true
• C. Both statements 1 and 2 are true
• D. Both statements 1 and 2 are false

Answer 1

Answer: (A)

$x^{x}(1+\log x)+a x^{a-1}+a^{x} \log _{e}^{a}$
$y=\frac{\log x}{\log 10}+\log x$
$\frac{d y}{d x}=\frac{1}{x \log 10}+\frac{1}{x}$