Reals are not equivalently as unrealizable or dubious as infinitesimals. Infintesimals can be constructively specified as nilpotent/nilsquare entities, numerical entities which when squared equal 0 but where those entities themselves aren't reducible to 0. All of this can be done in a constructive manner avoiding any use of infinity or classical logic that depends upon indirect proofs (ie excluded middle). John Bell's 'Primer of Infinitesimal Analysis' has good details.

The computational techniques from Automatic Differentiation use these types of entities to calculate derivatives exactly without approximating infinite (limiting) processes.

Calculus can be done constructively without infinite limiting processes purely algebraically using these nilpotents. And from a geometric interpretation there is nothing nonsensical about a tangent line to a curve.

Also you don't differentiate numbers (real or rational), you can only differentiate functions.

Also the idea of a actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.