Can someone help me solve this?
A point moves along the intersection of the elliptic paraboloid z=x^2+3y^2 and the plane x=2. At what rate is z changing with y when the point is at (2,1,7)?
The rate of change is simply the derivative!
The question asks for,
$\displaystyle \frac{ \partial z}{ \partial y } $
Which means we want the partial derivative with respect to y leaving x constant.
If $\displaystyle z = x^2+3y^2 $ then $\displaystyle \frac{ \partial z}{ \partial y } = 6y $
Sub in your given point and you have your answer.