Differentiation and Tangency. Definition. Suppose X and Y are normed vector spaces, A is a subset of X and f : A Y . Summary: Differentiation and Tangency. Definition. Suppose X and Y are normed vector spaces, A is a subset of X and f : A Y . For each v X we let vf = {(a, w) int A × Y : w = lim t0 1 t [f(a + tv) - f(a)]}. Owing to the uniqueness of limits we find that vf is a function with values in Y which we call the partial derivative of f with respect to v. In case X = Rn and i {1, . . . , n} we let if = ei f and call this function the i-th partial derivative of f; here ei is the i-th standard basis vector in Rn . Proposition. Suppose a int A. Then (i) a dmn 0f(a) and 0f(a) = 0; (ii) if v X, a dmn vf and c R then a dmn cvf and cvf(a) = cvf(a); (iii) if X = R then a dmn 1f(a) if and only if f is f is differentiable at a in which case f (a) = 1f(a). Proof. (i) and (iii) are immediate. To prove (ii) we suppose v X, a dmn vf,c R {0} and > 0. Choose > 0 such that Collections: Mathematics