Say we have a function $f(x,y)$, then the meaning of the partial derivative is clear: $$\lim _{dx \to 0}\frac{f(x+dx,y)-f(x,y)}{dx}$$
However, the total derivative is written as $$\frac{df}{dx}=\frac{\delta f}{\delta x}+ \frac{\delta f}{\delta y}\frac{dy}{dx}$$
But my confusion with this is that $y$ and $x$ are variables for the function $f$, and we can only take derivatives of functions, not derivatives of variables. So for $\frac{dy}{dx}$ to have any meaning, we have to interpret $y$ as a function, but what function? We never started out with a function $y(x)$, so how do we interpret $\frac{dy}{dx}$?