What is the differential of infinity?
Stringing words together doesn't always formulate a mathematical question. You have to define what your personal vocabulary is - because it isn't clear what you want to know. If you can't give definitions for your vocabulary, try giving a specific example that illustrates your question.
You seem to have several misunderstandings.
1. I am not sure if the following is the current convention. I think the convention is that infinity is NOT an acceptable value for a constant. It is defined as (a) the unbounded limit of an infinite series with values growing large and large without any finite limit, or (2) the limit of a continuous function, say f(x), as x approaches a value for which f(x) gets larger and larger without any finite limit.
2. Differentials are applied to functions. If you apply a differential to a function which is defined to be a constant, you get zero.
Separate names with a comma.