In Italian we have the notion of derivabilità and differenziabilità. In one dimension they are equivalent; in more dimensions, instead:
A function is derivabile if partial derivatives exist.
A function is differenziabile if it is also localmente linearizzabile (locally linearizable), that is, if $$\lim_{(x,y)\to(x_0,y_0)}\frac{f(x,y)-f(x_0,y_0)-\partial_xf(x_0,y_0)(x-x_0)-\partial_yf(x_0,y_0)(y-y_0)}{\sqrt[]{(x-x_0)^2+(y-y_0)^2}}=0$$
Is there an English term which translates differenziabile, other than locally linearizable?