 
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 ith partial derivative of f; here ei is the ith 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
