PX275 - A3a - the chain rule

consider the elevation $h (x, y)$ at a point P(x,y)
walk a distance, $s$ , to get to the point, $P$
height changes during the walk
how does the height vary during the walk?
movement along $x + y$ is given by $\frac{d x}{d s}$ and $\frac{d y}{d s}$
change in height is given in terms of $x$ and $y$ by the total differential:

\begin{aligned} d h & = \frac{\partial h}{\partial x} d x + \frac{\partial h}{\partial y} d y \\ d h & = \frac{\partial h}{\partial x} \frac{d x}{d s} d s + \frac{\partial h}{\partial y} \frac{d y}{d s} d s \\ ∴ \frac{d h}{d s} & = \frac{\partial h}{\partial x} \frac{d x}{d s} + \frac{\partial h}{\partial y} \frac{d y}{d s} \end{aligned}