The Chain Rule & Implicit Differentiation · Challenge problem

Let $f(x,y) = \dfrac{x^2y}{x^2 + y^2}$ for $(x,y) \ne (0,0)$ and $f(0,0) = 0$. Along the line $x = t$, $y = 2t$ put $g(t) = f(t, 2t)$. The chain rule would predict $g'(0) = f_x(0,0)\cdot 1 + f_y(0,0)\cdot 2$. Compute the actual $g'(0)$ minus this prediction.