462 CHAPTER 22. DERIVATIVE OF A FUNCTIONS OF MANY VARIABLES

13.5.4, the following computation shows the desired result.

|Tx|= |Ax|=

(m

∑j=1

∣∣∣(Ax) j

∣∣∣2)1/2

=

 m

∑j=1

∣∣∣∣∣ n

∑k=1

A jkxk

∣∣∣∣∣21/2

≤

 m

∑j=1

∣∣∣∣∣∣(

n

∑k=1

(A jk)2

)1/2( n

∑k=1

|xk|2)1/2

∣∣∣∣∣∣2

1/2

= |x|

 m

∑j=1

∣∣∣∣∣∣(

n

∑k=1

(A jk)2

)1/2∣∣∣∣∣∣2

1/2

Definition 22.2.3 Let f : U →Rp where U is an open set in Rn for n, p ≥ 1 and letx ∈U be given. Then f is defined to be differentiable at x ∈U if and only if there exists alinear transformation T such that,

f (x+h) = f (x)+Th+o(h) . (22.5)

The derivative of the function f, denoted by Df (x), is this linear transformation. Thus

f (x+h) = f (x)+Df (x)h+o(h)

If h= x−x0, this takes the form

f (x) = f (x0)+Df (x0)(x−x0)+o(x−x0)

If you deleted the o(x−x0) term and considered the function of x given by what isleft, this is called the linear approximation to the function at the point x0. In the case wherex ∈ R2 and f has values in R one can draw a picture to illustrate this.

Of course the first and most obvious question is whether the linear transformation isunique. Otherwise, the definition of the derivative Df (x) would not be well defined.

Theorem 22.2.4 Suppose f is differentiable, as given above in 22.5. Then T isuniquely determined. Furthermore, the matrix of T is the following p×n matrix(

∂f(x)∂x1

· · · ∂f(x)∂xn

)where

∂f

∂xi(x)≡ lim

h→0

f (x+tei)−f (x)

t,

the kth partial derivative of f .