Question

Second order derivative of $x^2$ with respect to $\ln x$ is

• A. $4x^2$
• B. $4x$
• C. $\frac{4x}{(\ln x{)}^2}$
• D. $\frac{4x-1}{(\ln x{)}^2}$

Answer 1

The correct option is A $4x^2$

Let $u=x^2$ and $v=\ln x$
$\Rightarrow \frac{du}{dv}=\frac{\frac{du}{dx}}{\frac{dv}{dx}}$
$\therefore \frac{du}{dv}=\frac{2x}{\frac{1}{x}}=2x^2$
$\Rightarrow \frac{d^{2}u}{d{v}^2}=\frac{\frac{d}{dx}\left(\frac{du}{dv}\right)}{\frac{dv}{dx}}$
$\therefore \frac{d^{2}u}{d{v}^2}=\frac{4x}{\frac{1}{x}}=4x^2$