There are two closely related interpretations of the derivative of a generating function in combinatorics.

• If $A(x) = \sum a_n x^n$ counts the number of $A$-structures on an $n$-element ordered set, then $A'(x) = \sum na_n x^{n-1}$ counts the number of ways to add a distinguished element to an $n-1$-element ordered set and to choose an $A$-structure on the result.

• If $A(x) = \sum \frac{a_n}{n!} x^n$ counts the number of $A$-structures on an $n$-element set, then $A'(x) = \sum \frac{a_n}{(n-1)!} x^{n-1}$ counts the number of ways to add an element to an $n-1$-element set and to choose an $A$-structure on the result.

Actually in combinatorics it is more natural (in both setups) to consider $x \frac{d}{dx}$, which is referred to as "pointing." This opens up a lot of interesting ideas; for example, one can interpret certain differential equations combinatorially and try to find their solutions symbolically.