Q:

Meaning of the phrase "the derivative is independent of the differential"

I was trying to read Riemann's "Foundations for a general theory of functions of a complex variable" and very early on ran into the phrase. 

A complex variable $w$ is said to be a function of another complex variable $z$, if $w$ varies with $z$ in such a way that the value of the derivative $\frac{dw}{dz}$ is independent of the value of the differential $dz$.

I haven't a clue what this might mean.

A:

Riemann is most likely not thinking of the differential $dz$ as tending anywhere, but rather adopting a Leibnizian attitude whereby one assigns an infinitesimal value to the $dz$, computes the corresponding change in $w$, forms a quotient and uses the law of homogeneity to attribute an assignable value to it. Then "being independent of the differential" is understood quite literally.

