In an x-y coordinates, $dy/dx$ is said to be the tangent to a curve at some point and giving us the slope there.
If this curve was parametrized by it's arc length $s$, then we might have $dy/ds$ and $dx/ds$ at some point.
How can we interpret $dy/ds$ and $dx/ds$, what do they give us?
Best Answer
Imagine a point travelling along the curve. Then $s$ is a distance travelled form some arbitrary $s=0$ point. Now, $dx/ds$ and $dy/ds$ are vector components in the $X$ and $Y$ axis directions of the point's movement. If the point travels with a speed $v=ds/dt$ then $v\cdot dx/ds$ is the 'horizontal' component of the point's velocity, and $v\cdot dy/ds$ – the 'vertical' one.