Question

If $y=x^2\log x$, prove that $\frac{d^{3}y}{d{x}^3}=\frac{2}{x}$

Answer 1

Consider $y=x^2\log x$

Differentiate with respect to $x$ we get

$\frac{dy}{dx}=2x\log x+x^2\cdot \frac{1}{x}$

$\Rightarrow \frac{dy}{dx}=2x\log x+x$

Again differentiate with respect to $x$ we get

$\Rightarrow \frac{d^{2}y}{d{x}^2}=2\log x+2x\cdot \frac{1}{x}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=2\log x+2$

Again differentiate with respect to $x$ we get

$\Rightarrow \frac{d^{3}y}{d{x}^3}=2\cdot \frac{1}{x}+0$

$\Rightarrow \frac{d^{3}y}{d{x}^3}=\frac{2}{x}$

Hence proved.