2nd order derivative - chain rule

Question

I'm facing this problem: let $g(t)=f(x(t),y(t))$ and $f(x,y)=2x^3y-y^2x+x+2y$. Compute $g''(0)$ knowing that $x(0)=y(0)=x'(0)=y'(0)=0$ and $x''(0)=y''(0)=-1$. I proceeded by computing the first derivative of $g(t)$ with the chain rule, that is $g'(t)=f_{x}(x(t),y(t))x'(t)+f_{y}(x(t),y(t))y'(t)=(6x^2y-y^2+1)x'(t)+(2x^3-2xy+2)y'(t)$ and by substituting the given value this expression is equal to zero. So i don't need to compute the second order derivative. Is it correct or am i missing something? Thanks a lot in adance for your answers.