Why it's that way
From this wonderful site Earliest Known Uses of Some of the Words of Mathematics. See especially the first paragraph I transferred below:
DERIVATIVE. Leibniz’s immediate successors, the Bernoullis and Euler, also wrote in Latin but by the middle of the 18th century French was becoming a major mathematical language. In modern English, when we speak of obtaining the derivative of a function by the process of differentiation, we are combining Lagrange’s French and Leibniz’s Latin.
Joseph Louis Lagrange (1736-1813) used derivée de la fonction and fonction derivée de la fonction as early as 1772 in “Sur une nouvelle espece de calcul relatif a la différentiation et a l'integration des quantités variables,” Nouveaux Memoires de l'Academie royale des Sciences etBelles-Lettres de Berlin. (Oeuvres, Vol. III) pp. 441-478. Lagrange states, for instance (first pages):
...on designe de même par u'' une fonction derivée de u' de la même maniére que u' l'est de u, et par u''' une fonction derivée de même de u'' et ainsi, ...
... les fonctions $u, u', u'', u''', u^{IV}$, ... derivent l'une de l'autre par une même loi de sorte qu'on pourra les trouver aisement par une meme operation répetée. [the functions $u, u', u'', u''', u^{IV}$, ... are derived one another from the same law, such that ...]
...
The term DERIVATIVE (from Lagrange) made occasional appearances in English texts of the 19th century: in 1834 W. R. Hamilton referred to the “derivative or differential coefficient” in his On a General Method in Dynamics Philosophical Transactions of the Royal Society. But it was not the favoured term. In his Differential and Integral Calculus (1891) George A. Osborne used the term differential coefficient but acknowledged, “The differential coefficient is sometimes called the derivative.” In the 20th century the preference has been reversed: see G. H. Hardy A Course of Pure Mathematics (1908, p. 197).
The reason that we use "derivative" the way we do is because of how it arrived in English, and that often leads to these puzzling inconsistencies in the way we use the words. It may well be that it enters other languages in alternative ways that allow you to use "derivable" in that language, but there is no reason to expect English to adopt the custom used by other languages just to make it consistent across languages.
(Probably) why English doesn't use "derivable"
I'm pretty much of the same feeling that Seth expressed: "derive" is a much more generic verb than "differentiate," and it doesn't seem sensible to look for excuses to use "derive" this way.
