Why is it legal to consider $\frac{dy}{dx}$ for a parametric curve when y may not be a valid function of x? When deriving the formula for the derivative of a parametric curve, in the form of $x = x(t)$ and $y = y(t)$, the chain rule is applied to $\frac{dy}{dt}$ to obtain $\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$, from which the slope of $y$ with respect to $x$ can be obtained.
My question is: why is it legal to use $\frac{dy}{dx}$ when $y$ is often not a function of $x$?
For example, the curve described by the parametric equations $x=6sin(t)$ and $y=t^2+t$ (image here) is clearly not a function of $x$, since it fails the vertical line test infinitely many times, and yet $\frac{dy}{dx} = \frac{2t+1}{6cos(t)}$. 
What is the intuition behind $\frac{dy}{dx}$ in this case? I usually think of $\frac{dy}{dx}$ as the (unique) slope induced from a small change in $x$, but that doesn't make sense here, since a small change in $x$ corresponds to infinitely many changes in $y$.

There is an important theorem in calculus which says, basically, that for any point on a (nice) curve where the curve is non-vertical and not self-intersecting, there is a piece of the curve surrounding that point where the curve looks like the graph of a function. In other words, there is a function $y(x)$ whose graph coincides with that small piece of the curve. Since derivatives only care about the immediate surroundings of any given point, this lets you make sense of $\frac{dy}{dx}$ at that point.
