A point moves along the intersection of the elliptic paraboloid $z=x^2+3y^2$, and the plane $y=1$. At what rate is $z$ changing with respect to $x$ when the point is at $(2,1,7)$.
My Work
$$\displaystyle{z=x^2+3y^2}\ \text{and} \ y=1 \ \text{and} \ P(2,1,7)$$ \begin{align} \frac{\delta z}{\delta x} &= 2x \\ \frac{\delta z}{\delta x}\Bigg\vert_{x=2} &= 4 \end{align} Is my answer and method correct, and if so could you please highlight how?
Your work seems fine to me. Basically when $y = 1$, the point $P$ is on a parabola given by $z = x^2+3$ and that this parabola is on the plane $y = 1$. Thus the rest is what you've done.