Let $f(x,y)$ is sufficiently smooth for all I write and let $x=x(t)$ and $y=y(t)$. I am curious about a proper notation of derivatives.
Given the limit definition of a partial derivative, I would say:
$$\frac{\partial f(x(t),y(t))}{\partial t}=0$$ and
$$\frac{{\rm d} f(x(t),y(t))}{{\rm d} t}=\frac{\partial f(x,y)}{\partial x}\frac{{\rm d} x(t)}{{\rm d} t}+\frac{\partial f(x,y)}{\partial y}\frac{{\rm d} y(t)}{{\rm d} t}$$
If the above is correct notation, why would many physicists (notably it started in thermodynamics) write:
$$\frac{\partial f(x(t),y(t))}{\partial t}=\frac{\partial f(x,y)}{\partial x}\frac{{\rm d} x(t)}{{\rm d} t}+\frac{\partial f(x,y)}{\partial y}\frac{{\rm d} y(t)}{{\rm d} t}$$
what would be the motivation to write the above instead of the total derivative? BTW (I also noticed a similar notation in the answer to this at this forum)