3.16 Change in a function along a curve

We wish to analyze how $f(x,y)$ changes as we move from $(x,y)$ to a nearby point $(x+h,y+k)$ in the plane. There are many possible routes, and we consider here movements parallel to the co–ordinates axes. We move horizontally from $(x,y)$ to $(x+h,y)$ and then move vertically from $(x+h,y)$ to $(x+h,y+k)$. Then the change in $f(x,y)$ is:

Hence all the subsequent chain rules will involve partial derivatives in the $x$ and $y$ directions. As we move along the curve $C$ with parametric form $(x(t),y(t))$, then $f(x(t),y(t))$ is a function of $t$.