It’s sometimes claimed that this notation is inappropriate in the context of the multivariable chain rule because, e.g., “If z(t) = z(x(t), y(t)), then dz/dt = dz/dx * dx/dt + dz/dy * dy/dt, which is very different from the dz/dx * dx/dt or dz/dy * dy/dt the notation would suggest! [For example, if x(t) = t, y(t) = ln(t), and z(x, y) = x[sup]3[/sup] + 5y, then dx/dt = 1, dy/dt = 1/t, dz/dx = 3x[sup]2[/sup], dz/dy = 5, and dz/dt = 3x[sup]2[/sup] + 5/t, not merely 3x[sup]2[/sup] or merely 5/t]”.
But the problem here is not in treating all these ratio-type things as ratios. That part is totally fine!
Rather, the problem is that “d” is overloaded in such phrasing to stand for multiple different difference operators, which our partial derivative notation unfortunately (and, I would agree, awfully) obfuscates: in the phrasing of the above quote, “dz/dt”, we are imagining differences as t varies (and x and y vary along with it, and z along with that), whereas in “dz/dx” we are imagining differences as x varies but y is held constant, and in “dz/dy” we are imagining differences as y varies but x is held constant. In the first of these, x and y are yoked together by their dependence on t, while in the latter two, x and y are taken to independently vary (and thus there is no such thing as the quantity t).
If we explicitly labeled our difference operators to indicate their “partiality”, there would be no problem. The various relevant difference operators for this problem would be related through the linearity principle that total change in z is the sum of the change from varying x alone and the change from varying y alone (this is the characteristic of “total” differentiability), from which the multivariable chain rule follows straightforwardly in precisely the way the fractional notation would accurately indicate.