The total differential is found by taking the limit of $$\Delta f \approx f_x \Delta x+ f_y \Delta y$$ as $\Delta x, \Delta y \rightarrow 0$ to give $$df=f_x dx +f_y dy$$ Is this equation not just saying that $$0=f_x. 0 +f_y.0$$
If not please can you explain why not?
Although both $\Delta x$ and $\Delta y\rightarrow 0$, this does not necessarily mean that the product $f_x \Delta x\rightarrow 0$. This is because, the $dx$ or $dy$ in question, are not actually zero, they approach zero but never reach it.
For example: $$\frac{\partial y}{\partial x}(x_0)=\lim_{h\rightarrow 0}{y(x_0+h)-y(x_0-h)\over 2h}$$
The space between the two $y$'s is never zero, but it heads torwards that while the denominator also heads towards zero! Eventually these find a sort of equilibrium, at $\frac{\partial y}{\partial x}$.