334 CHAPTER 9 CALCULUS

compute the values of the derivatives f(k) ( (^0) ), k = 1,2,3, .... (We are using the nota­
tion f(k) for the kth derivative of f.)
Partial Solution: At first you might guess that we can let n = 1/ x and get
1
f(x) = x2^1

• 
  

  1 -

  
  

   x2+ 1

  
  

  -+1
  x2
  for all x. The trouble with this is that it is only valid for those values of x for which
  l/x is an integer! So we know nothing at all about the behavior of f(x) except at the
  points x = 1,1/2, 1/3, ....
  But wait! The limit of the sequence 1, 1/2, 1/3, ... is 0, and the problem is only
  asking for the behavior of f(x) at x = O. So the strategy is clear: wishful thinking
  suggests that f(x) and its derivatives agree with the behavior of the function w(x) :=
  1/ (x^2 + 1) at x = O.
  In other words, we want to show that the function

  
  

v(x) := f(x) -w(x)

satisfies

V(k)(O) = (^0) , k = 1,2,3, ....
This isn't too hard to show, since v(x) is "almost" equal to 0 and gets more like 0 as x
approaches 0 from the right. More precisely, we have

o = v( 1) = v ( �) = v ( �) = ....

Since v(x) is continuous, this means that v(O) = O. Here's why: Let

Then lim Xn = 0 and

n-tOO

1 1

Xl = 1, X2 = 2"' X 3 = 3"' ....

lim v(xn) = lim 0 = (^0) ,
n-+oo n-i'OO
and v(x) = 0 by the definition of continuity (see page 323).
(5)
Now you complete the argument! Use Rolle's theorem to get information about
the derivative, as x ---t 0, etc.^9

A Useful Tool

We will conclude our discussion of differentiation with two examples that illustrate a

useful idea inspired by logarithmic differentiation.

(^9) See Example 9. 4. 3 on page 346 for a neat way to compute the derivatives of I/(r + I) at O.