Apologies if there is a duplicate somewhere; I couldn't find one.
The use of the root "deriv" in the context of differentiation seems odd: we have differentiation, differentials, differentiable, differential equations, and then for some odd reason, "derivative." Why/how did this happen?
On
In French there is less of a problem with terminology: differentiation is called "dérivation", and derivative is, you guessed it, "la dérivée". The term "fonction dérivée" was originally introduced by Lagrange. The English term "differentiate" ultimately derives from "differences"; namely, Leibniz originally studied finite differences and discovered certain patterns that led him to introduce (infinitesimal) differentials.
I believe the term "derivative" arises from the fact that it is another, different function $f'(x)$ which is implied by the first function $f(x)$. Thus we have derived one from the other. The terms differential, etc. have more reference to the actual mathematics going on when we derive one from the other.