Let $\vec y$ be a vector that represents the path of a particle through space. If we define $w$ as the length of the path, would it be correct to say that $\displaystyle \frac{d\vec y}{dw}$ evaluated at any point on the curve is equal to the unit vector tangent to the path at that point?
Much appreciated.
Yes, that's right. If we parametrize the curve with respect to some parameter $t$, then $\frac{d\vec y}{dt}$ at any given point is a vector that is tangent to the curve at any given point, so $\frac{\frac{d\vec y}{dt}}{\|\frac{d\vec y}{dt}\|}$ is the unit tangent vector. But in the case when we're parametrizing the curve with respect to the arc length $w$, the unit tangent vector is just $\frac{d\vec y}{dw}$, because $\|\frac{d\vec y}{dw}\|=1$. (Try proving that.)
EDIT: I should mention that the usual notation used is $\vec r$ and $s$ instead of $\vec y$ and $w$.