Question

Find the second order derivative of the given functions.

$x^{3}logx$

Answer 1

Let y = $x^{3}logx$ ,Differentiating twicely w.r.t. x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(x^{3}logx\right)=x^3\frac{d}{dx}\left(logx\right)+logx\frac{d}{dx}\left(x^3\right)=x^3\left(\frac{1}{x}\right)+\left(logx\right)\left(3x^2\right)=x^2(1+3logx)\left(Usingproductrule\right)and\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left\{x^2\right(1+3logx\left)\right\}=x^2(0+\frac{3}{x})+(1+3logx)\left(2x\right)=3x+2x(1+3logx)=x(5+6logx)$