Question

Derivative of ${{\log }_{10}}\,x$ with respect to ${{x}^{2}}$ is

• A. $2{{x}^{2}}\,{{\log }_{e}}\,10$
• B. $\frac{{{\log }_{10}}\,e}{2{{x}^{2}}}$
• C. $\frac{{{\log }_{e}}\,10}{2{{x}^{2}}}$
• D. ${{x}^{2}}\,{{\log }_{e}}\,10$

Answer 1

Answer: (B)

$\frac{{{\log }_{10}}\,e}{2{{x}^{2}}}$

Answer 2

Let $u={{\log }_{10}}x=\frac{{{\log }_{e}}x}{{{\log }_{e}}10}={{\log }_{10}}\,e\,{{\log }_{e}}x$
$\therefore$ $\frac{du}{dx}=\frac{{{\log }_{10}}\,e}{x}$
and $v={{x}^{2}}$
$\therefore$ $\frac{dv}{dx}=2x$
Now, $\frac{du}{dv}=\frac{{{\log }_{10}}e}{x+2x}=\frac{{{\log }_{10}}e}{2{{x}^{2}}}$