Derivatives

In this chapter we shall confine our attention to real functions defined on intervals or segments. This is not just a matter of convenience, since genuine differences appear when we pass from real functions to vector-valued ones.

THE DERIVATIVE OF A REAL FUNCTION

Definition 1. Let ƒ be defined (and real-valued) on [a,b]. For any x∈[a,b] form the quotient

(1) φ(t)=(ƒ(t)-ƒ(x))/(t-x) (a<t<b,t≠x),

and define

(2) (ƒ^′)(x)=(lim_t→x)(φ(t))

provided this limit exists in accordance with the definition.

We thus associate with the function ƒ a function (ƒ^′) whose domain is the set of points x at which the limit (2) exists; (ƒ^′) is called the derivative of ƒ.

If (ƒ^′) is defined at a point x, we say that ƒ is differentiable at x. If (ƒ^′) is defined at every point of a set E⊂[a,b], we say that fis differentiable on E.

It is possible to consider right-hand and left-hand limits in (2); this leads to the definition of right-hand and left-hand derivatives. In particular, at the endpoints a and b, the derivative, if it exists, is a right-hand or left-hand derivative, respectively. We shall not, however, discuss one-sided derivatives in any detail.

If ƒ is defined on a segment (a,b) and if a<x<b, then (ƒ^′)(x) is defined by (1) and (2), as above. But (ƒ^′)(a) and (ƒ^′)(b) are not defined in this case.

Theorem 1. Let ƒ be defined on [a, b]. If fis differentiable at a point x & [a, b],

then f is continuous at x.

Proof As t—.x, we have, by Theorem 4.4,

F(t) —f@) ,

ft) - 40) =7P=™ 6 y7@)-0=0

The converse of this theorem is not true. It is easy to construct continuous

functions which fail to be differentiable at isolated points. In Chap. 7 we shall

even become acquainted with a function which is continuous on the whole line

without being differentiable at any point!

5.3. Theorem Suppose f and g are defined on [a, b] and are differentiable at a

point x & [a,b]. Thenf + g, fg, and f/q are differentiable at x, and

@) S49 =f@ +9);

(6) (f9)'(™) =f’ @)9@) + F()9'@):

© (Z ) (x) = IO)F re CFO).

In (c), we assume of course that g(x) ¥ 0.

Proof (a) is clear, by Theorem 4.4. Leth =/g. Then

ht) — hO) =fOLIO) — 9@)] + gL — SO)

DIFFERENTIATION 105

If we divide this by ¢ — x and note that f(t) > f(x) as t > x (Theorem 5.2),

(6) follows. Next, let h =f/g. Then

ht)—h(x)_ 1 f(t) -#@) gt) — 9)

tox — g(t)g(x) [a9 tax SOI y I

Letting t > x, and applying Theorems 4.4 and 5.2, we obtain (c).

5.4 Examples The derivative of any constant is clearly zero. If fis defined

by f(x) = x, then f’(x) = 1. Repeated application of (5) and (c) then shows that

x" is differentiable, and that its derivative is nx"~1, for any integer n (if n <0,

we have to restrict ourselves to x #0). Thus every polynomial is differentiable,

and so is every rational function, except at the points where the denominator is

zero.

The following theorem is known as the ‘“‘chain rule’? for differentiation.

It deals with differentiation of composite functions and is probably the most

important theorem about derivatives. We shall meet more general versions of it

in Chap. 9.

5.5 Theorem Suppose f is continuous on [a, bl, f'(x) exists at some point

x € [a,b], g is defined on an interval I which contains the range of f, and g is

differentiable at the point f(x). If

At)=9F)) (ast<b),

then h is differentiable at x, and

(3) W(x) = oF) F'@).

Proof Let y =/(x). By the definition of the derivative, we have

(4) ft) ~f) = @ — LF’) + ue),

(5) HS) — gy) = (5 — VI9’Q) + (5),

where fe [a, 5], s e/, and u(t) +0 as t+ x, o(s) 20 assoy. Lets =f(t).

Using first (5) and then (4), we obtain

A(t) — h(x) = 9 f(t) — 9S)

= 40) —£@))- [9’O) + 0)]

= (t— x): [f'@) +ut)] -[9O) + 0),

or, if t # x,

ht) — A(x) _

t—-x

Letting t+ x, we see that sy, by the continuity of f, so that the right

side of (6) tends to g’(y)/’(x), which gives (3).

(6) [g'(y) + os) [F'°@) + u(t].

106 PRINCIPLES OF MATHEMATICAL ANALYSIS

5.6 Examples

(a) Let f be defined by

|

Q) f= x sin 5 (x #0),

0 (x = 0).

Taking for granted that the derivative of sin x is cos x (we shall

discuss the trigonometric functions in Chap. 8), we can apply Theorems

5.3 and 5.5 whenever x #0, and obtain

(x) = sin — Leos b

(8) tf (x) = sin = 78x (x #0).

At x = 0, these theorems do not apply any longer, since 1/x is not defined

there, and we appeal directly to the definition: for ¢ #0,

fO)-fO_ 1

7x0 sin =

As t+ 0, this does not tend to any limit, so that £’(0) does not exist.

(5) Let f be defined by

2 sin

0) foo = x* sin x (x #0),

0 (x = 0),

As above, we obtain

| 1

(10) Sx) = 2x sin 77 cos x (x #0).

At x =0, we appeal to the definition, and obtain

F@) ~ FO)

tsin |

t-0

t

Slt] (#0);

letting t > 0, we see that

(11) f'(0) =0.

Thus / is differentiable at all points x, but /’ is not a continuous

function, since cos (1/x) in (10) does not tend to a limit as x + 0.

DIFFERENTIATION 107

MEAN VALUE THEOREMS

5.7 Definition Let f be a real function defined on a metric space ¥. We say

that fhas a local maximum at a point p e X if there exists 6 > 0 such that f(g) <

S(p) for ali ¢g e X with d(p, g) < 6.

Local minima are defined likewise.

Our next theorem is the basis of many applications of differentiation.

5.8 Theorem Let f be defined on [a, b); if f has a local maximum at a point

x € (a, b), and if f'(x) exists, then f"(x) = 0.

The analogous statement for local minima is of course also true.

Proof Choose 6 in accordance with Definition 5.7, so that

a<x-—d<x<x+d<5.

Ifx—d<t<-x, then

fO-1@), 5

t-—x

Letting 1 > x, we see that f’(x) > 0.

Ifx<t<x+ 6, then

fO-1@) <9

t-x

which shows that f’(x) < 0. Hence f’(x) = 0.

5.9 Theorem If f and g are continuous real functions on [a,b] which are

differentiable in (a, b), then there is a point x 6 (a, b) at which

[f(6) —F(@lg"(x) = [9) — g@) Lf’).

Note that differentiability is not required at the endpoints.

Proof Put

At) = [F) -S@ lO) -I9@-9@lft) @st<d).

Then A is continuous on [a, b], / is differentiable in (a, 5), and

(12) h(a) = f(0)g(a) — F(a)g(b) = hb).

To prove the theorem, we have to show that h’(x) = 0 for some x € (a, b).

If A is constant, this holds for every x ¢ (a, 5). If A(t) > h(a) for

some ¢ € (a, 5), let x be a point on [a, b] at which A attains its maximum

108 PRINCIPLES OF MATHEMATICAL ANALYSIS

(Theorem 4.16). By (12), x € (a, 6), and Theorem 5.8 shows that h’(x) = 0.

If h(t) < h(a) for some ¢ € (a, 5), the same argument applies if we choose

for x a point on [a, d] where A attains its minimum.

This theorem is often called a generalized mean value theorem; the following

special case is usually referred to as “‘the’’ mean value theorem:

§.10 Theorem If fis a real continuous function on [a, b] which is differentiable

in (a, b), then there is a point x é (a, b) at which

f(6) —f(@) = (6 — af").

Proof Take g(x) = x in Theorem 5.9.

5.11 Theorem Suppose f is differentiable in (a, b).

(a) Iff'(x) 20 for all x € (a, b), then f is monotonically increasing.

(b) If f'(x) =0 for all x € (a, b), then f is constant.

(c) Iff'(x) <0 for all x € (a, b), then f is monotonically decreasing.

Proof All conclusions can be read off from the equation

F(%2) — F%1) = 2 — x)F'@),

which is valid, for each pair of numbers x,, x, in (a, b), for some x between

x, and x.

THE CONTINUITY OF DERIVATIVES

We have already seen [Example 5.6(5)] that a function f may have a derivative

f' which exists at every point, but is discontinuous at some point. However, not

every function is a derivative. In particular, derivatives which exist at every

point of an interval have one important property in common with functions

which are continuous on an interval: Intermediate values are assumed (compare

Theorem 4.23). The precise statement follows.

5.12 Theorem Suppose f is a real differentiable function on {a, b] and suppose

S'(@ <4<f'(b). Then there is a point x € (a, b) such that f'(x) = 4.

A similar result holds of course if f’(a) > f’(b).

Proof Put g(t) = f(t) — At. Then g’(a) < 0, so that g(t,) < g(a) for some

t, € (a, 5), and g’(b) > 0, so that g(t.) < g(b) for some ¢, (a, b). Hence

g attains its minimum on [a, b] (Theorem 4.16) at some point x such that

a<x<b. By Theorem 5.8, g(x) =0. Hence /’(x) = 4.

DIFFERENTIATION 109

Corollary Jf f is differentiable on [a,b], then f’ cannot have any simple dis-

continuities on [a, b].

But f’ may very well have discontinuities of the second kind.

L’HOSPITAL’S RULE

The following theorem is frequently useful in the evaluation of limits.

5.13 Theorem Suppose f and g are real and differentiable in (a, b), and g'(x) #0

for all x & (a, b), where -0 Sa<b< +0. Suppose

(13) yee, asx—-a

if

(14) I(x) 0 and g(x) +0 as xa,

or if

(15) g(x) > +00 asx—a,

then

(16) DOs Aas xa

The analogous statement is of course also true if x > 4, or if g(x) > — 0

in (15). Let us note that we now use the limit concept in the extended sense of

Definition 4.33.

Proof We first consider the case in which —o <A < +00. Choose a

real number ¢ such that A <q, and then choose r such that A<r<g.

By (13) there is a point ¢ € (a, 6) such that a < x < ¢ implies

(17) LO) -,.

g(x)

Ifa<x<y<ce, then Theorem 5.9 shows that there is a point ¢ € (x, y)

such that

f@)-f0)_fO

(18) a <r,

ax)—9) gO"

Suppose (14) holds. Letting x — a in (18), we see that

(19) £9 cr eg (a<y<c).

gy)

110 PRINCIPLES OF MATHEMATICAL ANALYSIS

Next, suppose (15) holds. Keeping y fixed in (18), we can choose

a point c, € (a, y) such that g(x) > g(y) and g(x) > Oifa<x<c,. Multi-

plying (18) by [g(x) — g(v)V/g(x), we obtain

£0) - QD, {M

(20) re) < ra + 7) (a<x<¢).

If we let xa in (20), (15) shows that there is a point cz € (a, c,)

such that

NAGS)

(21) a) <q (a<x <4).

Summing up, (19) and (21) show that for any q, subject only to the

condition A < q, there is a point c, such that f(x)/g(x) <q ifa<x<c,.

In the same manner, if —0oo < A < +, and p is chosen so that

p <A, we can find a point c; such that

fa)

g(x)

and (16) follows from these two statements.

(22) (a<x<c3),

DERIVATIVES OF HIGHER ORDER

5.14 Definition If fhas a derivative /’ on an interval, and iff’ is itself differen-

tiable, we denote the derivative of f’ by f” and call f” the second derivative of f-

Continuing in this manner, we obtain functions

LL I IO», sf,

each of which is the derivative of the preceding one. f) is called the nth deriva-

tive, or the derivative of order n, of f.

In order for f (x) to exist at a point x, f~)) (7) must exist in a neighbor-

hood of x (or in a one-sided neighborhood, if x is an endpoint of the interval

on which f is defined), and f°~) must be differentiable at x. Since f"~) must

exist in a neighborhood of x, f‘"~ ?) must be differentiable in that neighborhood.

TAYLOR’S THEOREM

5.15 Theorem Suppose f is a real function on [a, b], n is a positive integer,

f"-» is continuous on [a, b], f(t) exists for every t ¢ (a, b). Let a, B be distinct

points of {a, b), and define

nn £th)

(23) rn) ="5

k=

(t — a).