Data Structure Differentiation, Intuition Building

Question

According to this paper differentiation works on data structures.

According to this answer:

Differentiation, the derivative of a data type D (given as D') is the type of D-structures with a single “hole”, that is, a distinguished location not containing any data. That amazingly satisfy the same rules as for differentiation in calculus.

The rules are:

 1 = 0
 X′ = 1
 (F + G)′ = F' + G′
 (F • G)′ = F • G′ + F′ • G
 (F ◦ G)′ = (F′ ◦ G) • G′

The referenced paper is a bit too complex for me to get an intuition. What does this this mean in practice? A concrete example would be fantastic.