Question

The derivative of ${\log }_{10}x$ with respect to $x^2$ is

• A. $\frac{1}{2x^2}{\log }_{e}10$
• B. $\frac{2}{x^2}{\log }_{10}e$
• C. $\frac{1}{2x^2}{\log }_{10}e$
• D. none of these

Answer 1

The correct option is C $\frac{1}{2x^2}{\log }_{10}e$

$\frac{d\left(lo{g}_{10}\right(x\left)\right)}{d{x}^2}$
$=\frac{d\left(\frac{lo{g}_{10}x}{dx}\right)}{\frac{d{x}^2}{dx}}$
$=\frac{d\left(\frac{\frac{lnx}{ln10}}{dx}\right)}{\frac{d{x}^2}{dx}}$
$=\frac{\frac{1}{ln10.x}}{2x}$
$=\frac{lne}{2x^{2}ln10}$
$=\frac{lo{g}_{10}e}{2x^2}$