466 CHAPTER 22. DERIVATIVE OF A FUNCTIONS OF MANY VARIABLES
Example 22.2.11 Let f (x,y) =(
x3y+ y2
xy2 +1
). Find Df (x,y) .
Simple computations show that the matrix of this linear transformation is
Df (x,y) =(
f1x (x,y) f1y (x,y)f2x (x,y) f2y (x,y)
)=
(3x2y x3 +2yy2 2xy
)provided the function is differentiable. It is left as an exercise to verify that this does indeedserve as the derivative. A little later, a theorem is given which shows that, since the functionis a C1 function, it is indeed differentiable.
Example 22.2.12 Consider the open set O in the space of p× p matrices consisting ofthose which have an inverse. Let φ (A)≡ det(A) . Then have a look at Problem 39 on Page430 to see a description of Dφ (F).
22.3 Exercises1. Determine which of the following functions are o(h).
(a) h2
(b) hsin(h)
(c) |h|3/2 ln(|h|)(d) h2x+ yh3
(e) sin(h2)
(f) sin(h)
(g) xhsin(√
|h|)+ x5h2
(h) exp(−1/ |h|2
)2. Here are some scalar valued functions of several variables. Determine which of these
functions are o(v). Here v is a vector in Rn, v = (v1, · · · ,vn).
(a) v1v2
(b) v2 sin(v1)
(c) v21 + v2
(d) v2 sin(v1 + v2)
(e) v1 (v1 + v2 + xv3)
(f) (ev1 −1− v1)
(g) (x ·v) |v|
3. Here are some vector valued functions of v ∈ Rn. Determine which ones are o(v).
(a) (x ·v)v
(b) sin(v1)v
(c)√|(x ·v)| |v|2/3
(d)√|(x ·v)| |v|1/2
(e)(
sin(√
|x ·v|)−√|x ·v|
)·
|v|−1/4
(f) exp(−1/ |v|2
)(g) vT Av where A is an n×n matrix.
4. Show that if f (x) = o(x), then f ′ (0) = 0.
5. Show that if limh→0 f (x) = 0 then x f (x) = o(x).
6. Show that if f ′ (0) exists and f (0) = 0, then f (|x|p) = o(x) whenever p > 1.