302 CHAPTER 17. THE DERIVATIVE OF A FUNCTION OF MANY VARIABLES

Recall from theorem 8.3.2 this shows the matrix of the linear transformation is as claimed.■

Other notations which are often used for this matrix or the linear transformation aref ′ (x) ,J (x), and even ∂f

∂xor df

dx . Also, the above definition can now be written in the form

f (x+v) = f (x)+p

∑j=1

∂f (x)

∂x jv j +o(v)

orf (x+v)−f (x) =

(∂f(x)

∂x1· · · ∂f(x)

∂xn

)v+o(v)

Here is an example of a scalar valued nonlinear function.

Example 17.3.4 Suppose f (x,y) =√

xy. Find the approximate change in f if x goes from1 to 1.01 and y goes from 4 to 3.99.

We can do this by noting that

f (1.01,3.99)− f (1,4) ≈ fx (1,2)(.01)+ fy (1,2)(−.01)

= 1(.01)+14(−.01) = 7.5×10−3.

Of course the exact value is√(1.01)(3.99)−

√4 = 7.4610831×10−3.

Notation 17.3.5 When f is a scalar valued function of n variables, the following is oftenwritten to express the idea that a small change in f due to small changes in the variablescan be expressed in the form

d f (x) = fx1 (x)dx1 + · · ·+ fxn (x)dxn

where the small change in xi is denoted as dxi. As explained above, d f is the approximatechange in the function f . Sometimes d f is referred to as the differential of f .

Let f : U → Rq where U is an open subset of Rp and f is differentiable. It was justshown that

f (x+v) = f (x)+(

∂f(x)∂x1

· · · ∂f(x)∂xp

)v+o(v) .

Taking the ith coordinate of the above equation yields

fi (x+v) = fi (x)+p

∑j=1

∂ fi (x)

∂x jv j +o(v) ,

and it follows that the term with a sum is nothing more than the ith component of J (x)vwhere J (x) is the q× p matrix

∂ f1∂x1

∂ f1∂x2

· · · ∂ f1∂xp

∂ f2∂x1

∂ f2∂x2

· · · ∂ f2∂xp

......

. . ....

∂ fq∂x1

∂ fq∂x2

· · · ∂ fq∂xp

 .