So we want to differentiate y =  4x²ln(x) with respect to y. For this we need to use the product rule.

The product rule is D {f(x)g(x)} = f(x)g'(x) + g'(x)f(x)

We can therefore make f(x) = 4x² and g(x) = ln (x)

f'(x) = 8x nad g'(x) = 1/x

Therefore dy/dx = 8xln(x) + 4x²/x which can be simpliefied to 8xln(x) + 4x, which can be further simplified to get the answer:

4x(2ln(x) + 1)