Answer
$$
6
$$
Work Step by Step
We are given that
\[
2=x \text { and } x^{2}+3 y^{2}=z
\]
Thus, now obtaining the intersection point of both curves
\[
\begin{array}{l}
2^{2}+3 y^{2}=z \\
4+3 y^{2}=z
\end{array}
\]
Differentiating with respect to x:
\[
\frac{d}{d y}\left(4+3 y^{2}\right)=\frac{d z}{d y}
\]
\[
6 y=\frac{d z}{d y}
\]
Rate of change of $z$ with respect to $y$ at the point (2,1,7)
\[
\begin{array}{l}
\left.\frac{d z}{d y}\right|_{y=1}=6(1) \\
\left.\frac{d z}{d y}\right|_{y=1}=6
\end{array}
\]
