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§2.3 Definition of Derivatives and the Matrix Representation of Derivatives 37

2.3 Definition of Derivatives and the Matrix Represen-
tation of Derivatives

Definition 2.15. Let U Ď Rm be an open set. A function f : U Ñ Rm is said to be
differentiable at x0 P A if there is a linear transformation from Rn to Rm, denoted by
(Df )(x0) and called the derivative of f at x0, such that

lim ››f (x) ´ f (x0) ´ (Df )(x0)(x ´ x0)››Rm = 0 ,
xÑx0 }x ´ x0}Rn

where (Df )(x0)(x ´ x0) denotes the value of the linear transformation (Df )(x0) applied to
the vector x ´ x0. In other words, f is differentiable at x0 P U if there exists L P B(Rn, Rm)
such that

@ ε ą 0, D δ ą 0 Q }f (x) ´ f (x0) ´ L(x ´ x0)}Rm ď ε}x ´ x0}Rn whenever }x ´ x0}Rn ă δ .

If f is differentiable at each point of U, we say that f is differentiable on U.

Example 2.16. Let L : Rn Ñ Rm be a linear transformation; that is, there is a matrix
[L]mˆn such that L(x) = [L]mˆn[x]n for all x P Rn. Then L is differentiable. In fact,
(DL)(x0) = L for all x0 P X since

lim }Lx ´ Lx0 ´ L(x ´ x0)}Rm = 0 .
xÑx0 }x ´ x0}Rn

Example 2.17. Let f : R2 Ñ R be given by f (x, y) = x2+2y. Define L(a,b)(x, y) = 2ax+2y.
Then L(a,b) is a linear transformation (from R2 to R) and

ˇˇx2 + 2y ´ a2 ´ 2b ´ L(a,b)(x ´ a, y ´ b)ˇˇ

a(x ´ a)2 + (y ´ b)2

ˇˇx2 + 2y ´ a2 ´ 2b ´ 2a(x ´ a) ´ 2(y ´ b)ˇˇ
= a(x ´ a)2 + (y ´ b)2

= (x ´ a)2 ´ b)2 ď |x ´ a| ;
a(x ´ a)2 + (y

thus ˇˇx2 ´ a2 ´ 2b ´ L(a,b)(x ´ ´ b)ˇˇ
a(x ´ a)2 + (y ´ b)2
lim + 2y a, y = 0.

(x,y)Ñ(a,b)

Therefore, f is differentiable at (a, b) and (Df )(a, b) = L(a,b).