460 CHAPTER 22. DERIVATIVE OF A FUNCTIONS OF MANY VARIABLES
Proof: From the above observation it follows that if f ′ (x) does exist, then 22.2 holds.Suppose then that 22.2 is true. Then
f (x+h)− f (x)h
− p =o(h)
h.
Taking a limit, you see that
p = limh→0
f (x+h)− f (x)h
and that in fact this limit exists which shows that p = f ′ (x).This theorem shows that one way to define f ′ (x) is as the number p, if there is one,
which has the property that
f (x+h) = f (x)+ ph+o(h) .
You should think of p as the linear transformation resulting from multiplication by the 1×1matrix (p).
Example 22.1.4 Let f (x) = x3. Find f ′ (x).
f (x+h) = (x+h)3 = x3 +3x2h+3xh2 +h3 = f (x)+3x2h+(3xh+h2)h.
Since(3xh+h2
)h = o(h), it follows f ′ (x) = 3x2.
Example 22.1.5 Let f (x) = sin(x). Find f ′ (x). f (x+h)− f (x) =
sin(x+h)− sin(x) = sin(x)cos(h)+ cos(x)sin(h)− sin(x)
= cos(x)sin(h)+ sin(x)(cos(h)−1)
hh
= cos(x)h+ cos(x)(sin(h)−h)
hh+ sinx
(cos(h)−1)h
h.
Now
cos(x)(sin(h)−h)
hh+ sinx
(cos(h)−1)h
h = o(h) . (22.3)
Remember the fundamental limits which allowed you to find the derivative of sin(x) were
limh→0
sin(h)h
= 1, limh→0
cos(h)−1h
= 0. (22.4)
These same limits are what is needed to verify 22.3.
How can you tell whether a function of two variables (u,v) is o
(uv
)? In general,
there is no substitute for the definition, but you can often identify this property by observingthat the expression involves only “higher order terms”. These are terms like u2v,uv,v4, etc.If you sum the exponents on the u and the v you get something larger than 1. For example,∣∣∣∣ vu√
u2 + v2
∣∣∣∣≤ 12(u2 + v2) 1√
u2 + v2=
12
√u2 + v2