Are both interpretations of the differential accurate?

Question

A differential can be written, informally, as the form of $du=\dfrac{\partial f}{\partial x_1}dx_1+\dfrac{\partial f}{\partial x_2}dx_2+...\dfrac{\partial f}{\partial x_n}dx_n$. In the textbook I am reading it states the differential can be interpreted in two ways. The first is that if we take $dx_1,...,dx_n$ as infinitesimal increments in the independent variables $x_1,...,x_n$ of a function $f$, then $du$ is the corresponding infinitesimal increment in the dependent variable $u$. The second is that we think of $dx_1,...,dx_n$ as finite increments, and $du$ is the corresponding increment in the value $u$, not on the (hyper)surface, but on its tangent (hyper)plane. In other words, it is the linear approximation to the increment in the function $f$. My question is, which is interpretation is valid? Or are both valid, but just in different contexts.

Answer 1

Both interpretations are certainly helpful. The first of them, talking about "infinitesimal increments", cannot be "accurate", and would not be acceptable in a formal argument. The second is nearer to the formal definition of differential, but does not encompass its full content. The latter describes precisely the idea of "linear approximation" in terms of a certain limit.