Question

Find the second order derivatives of the function
$x^3logx$

Answer 1

The correct answer is $=x(5+6\,logx)$
Let $y=x^3logx$
Then,
$\frac{dy}{dx}=\frac{d}{dx}(x^3logx)=logx.\frac{d}{dx}(x^3)+x^3.\frac{d}{dx}(logx)$
$=logx.3x^2+x^3.\frac{1}{x}=logx.3x^2+x^2$
$=x^2(1+3logx)$
$∴\frac{d^2y}{dx^2}=\frac{d}{dx}[x^2(1+3logx)]$
$=(1+3logx).\frac{d}{dx}(x^2)+x^2\frac{d}{dx}(1+3logx)$
$=(1+3logx).2x+x^2.\frac{3}{x}$
$=2x+6x\,logx+3x$
$=5x+6x\,logx$
$=x(5+6\,logx)$