Question

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

• 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)

$x^x(1+\log x)+a{x}^{a-1}+a^x{\log }_e^a$
$y=\frac{\log x}{\log 10}+\log x$
$\frac{dy}{dx}=\frac{1}{x\log 10}+\frac{1}{x}$