Question

Find the second order derivative of the following function:
$x^3\log x$

Answer 1

Let $y=x^{3}logx$. Then,
$\frac{dy}{dx}=logx\frac{d}{dx}\left(x^3\right)+x^3\frac{d}{dx}\left(logx\right)$
$\Rightarrow \frac{dy}{dx}=3x^{2}logx+x^3\times \frac{1}{x}$
$\Rightarrow \frac{dy}{dx}=x^2(3logx+1)$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=(3logx+1)\frac{d}{dx}\left(x^2\right)+x^2\frac{d}{dx}(3logx+1)$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=2x(3logx+1)+x^2\times \frac{3}{x}=x(6logx+5)$