The second order derivative of $y=x^4 \log x$ is:

$x^2(7+12 \log x)$

The correct answer is Option (2) → $x^2(7+12 \log x)$

$y = x^4 \log x$

$\frac{dy}{dx} = x^4 \cdot \frac{1}{x} + \log x \cdot 4x^3$

$= x^3 + 4x^3 \log x = x^3(1 + 4\log x)$

$\frac{d^2y}{dx^2} = 3x^2(1 + 4\log x) + x^3 \cdot \frac{4}{x}$

$= 3x^2 + 12x^2 \log x + 4x^2$

$= x^2(7 + 12\log x)$

$\frac{d^2y}{dx^2} = x^2(7 + 12\log x)$