You started out ok, but your second line is incorrect.

This part is right 4(3xy^2+1)^3. Now you must multiply by the derivative of the inside function.

So d/dx (3xy^2+1)= 6xy(dy/dx)+3y^2.

Putting the LHS together it's 4(3xy^2+1)^3 * (6xy(dy/dx)+3y^2)

The RHS of your work is correct.