A particle moves along the intersection of the elliptic paraboloid $z = x^2 + 2y^2$ and the plane $x = 2$. At the moment when the particle is at $(2, 1, 5)$, what is the rate of change of $z$ with respect to $y$?
Attempt: This question confuses me. I thought of using the directional derivative, but since they are asking just for the rate of change with respect to $y$ only I suppose taking a simple partial derivative of $z$ with respect to $y$ is sufficient: $$z=f(x,y),\;f_y=4y$$ $$f_y(2,1,5)=4$$
So thats the rate of change of $z$ with respect to $y$ at $(2,1,5)$. However, it is suspiciously easy and I think I am doing it wrong. Help please.