4.1.1 Definition. A real-valued functionfdefined in a neighborhood ofa∈R is said to bedifferentiable ataif the limit
f0(a) =Df(a) = df dx
a:= lim
x→a
f(x)−f(a) x−a = lim
h→0
f(a+h)−f(a) h
exists inR. The limit is then called thederivative off ata. Iff is differentiable at each member of a setE, thenf is said to be differentiable on E and the function
f0=Df = df dx
is called thederivative off on E. Iff0 is continuous onE, thenf is said to
becontinuously differentiable onE. ♦
It follows immediately from the definition that the derivative of a constant function is 0. Here are some nontrivial examples.
4.1.2 Example. We prove the following special cases of the power rule (the general power rule will be proved later): Letn∈Nandr=nor 1/n. Then
Dxr=rxr−1. (In the second casex6= 0, andx >0 ifnis even.)
The case r=nis obtained by lettingh→0 in the identity (x+h)n−xn
h =
n
X
j=1
(x+h)n−jxj−1
73
74 A Course in Real Analysis
(Exercise 1.2.4.) Each term in the sum tends toxn−1, and since there aren terms the formula follows.
For the caser= 1/nwe use the identity (x+h)1/n−x1/n
h = n
X
j=1
(x+h)1−j/nx(j−1)/n −1
(Exercise 1.4.15). As h→0, the term in square brackets tends to nx1−1/n, verifying the formula.
♦ For the next example, and indeed for the remainder of the book, we shall use the standard definitions of cosine and sine as coordinates of points on the unit circle.1 From this one can derive the usual trigonometric identities, which we shall invoke as needed.
h
cosh sinh
tanh h
1
1
FIGURE 4.1: sinh < h <tanh.
4.1.3 Example. Dsinx= cosx.From the identity sin2h+ cos2h= 1 and the inequalities
sinh < h <tanh, 0< h < π/2,
which may be derived with the help of Figure 4.1, we see that p1−h2<p
1−sin2h= cosh < sinh
h <1, 0< h < π/2. (4.1) Since sin(−h) =−sinhand cos(−h) = cosh, (4.1) holds for−π/2< h <0 as well. By the squeeze principle,
h→0limcosh= lim
h→0
sinh h = 1. From this and the calculation
cosh−1
h = cos2h−1 h(cosh+ 1) =−
sinh h
2
h (cosh+ 1)
1A more rigorous approach to the calculus of trigonometric functions may be based on the inverse sine function. This approach is described briefly in Section 4.4.
Differentiation onR 75 we see that
h→0lim
cosh−1 h = 0. Therefore,
sin(x+h)−sinx
h = sinxcosh+ cosxsinh−sinx h
= cosh−1
h sinx+sinh h cosx
→cosx as h→0. ♦
It is occasionally necessary to consider one-sided derivatives, which are defined by using one-sided limits in 4.1.1. Specifically, theleft-hand andright- hand derivatives are, respectively,
D`f(a) =f`0(a) := lim
x→a−
f(x)−f(a) x−a and Drf(a) =fr0(a) := lim
x→a+
f(x)−f(a) x−a .
From the general theory of limits, a function is differentiable ataiff it has equal right-hand and left-hand derivatives at a. For example, atx= 0 the functionf(x) =|x|has right-hand derivative 1 and left-hand derivative −1 and so is not differentiable there.
Although we shall have no need to do so, one may even consider the more general expressions
lim infx→a
x∈E
f(x)−f(a)
x−a and lim sup
x→ax∈E
f(x)−f(a) x−a ,
where a is an accumulation point of E. The so-called Dini derivates are obtained by takingE to be intervals of the form (c, a) and (a, c).
The following proposition provides a useful characterization of differentia- bility. It asserts that forxneara,f(x) is approximated by the linear function y=f(a) +f0(a)(x−a), the equation of the tangent line at a.
4.1.4 Proposition. Letf be defined in a neighborhoodN(a)of a. Thenf is differentiable ataiff there exists a function η onN(a), continuous at a, such that
f(x) =f(a) +η(x)(x−a) for all x∈ N(a). In this case, f0(a) =η(a).
Proof. If such a functionη exists, then f(x)−f(a)
x−a =η(x)→η(a) asx→a,
76 A Course in Real Analysis
hencef0(a) exists and equalsη(a). Conversely, iff is differentiable ata, define
η(x) =
f(x)−f(a)
x−a ifx∈ N(a)\ {a}, f0(a) ifx=a.
Thenη has the required properties.
4.1.5 Corollary. If f is differentiable ata, thenf is continuous there.
Proof. Simply note thatf(x) =f(a) +η(x)(x−a)→f(a) asx→a.
The example|x|considered above shows that the converse of the corollary is false:|x|is continuous at 0 but not differentiable there. It is a remarkable fact that there are continuous functions onRthat arenowhere differentiable (see 8.9.7).
4.1.6 Theorem. Ifc∈Randf and g are differentiablea, then so aref+g, cf,f g, andf /g, the last provided thatg(a)6= 0. Moreover, in this case, (a) (f+g)0(a) =f0(a) +g0(a), (b) (cf)0(a) =cf0(a),
(c) (f g)0(a) =f(a)g0(a) +f0(a)g(a), (d)f g
0
(a) = g(a)f0(a)−f(a)g0(a) g2(a) . Proof. We prove only (d). Leth=f /g. Sincegis continuous ataandg(a)6= 0, his defined in a neighborhoodN(a) on whichg is not 0. Forx∈ N(a)\ {a}, a little algebra shows that
h(x)−h(a)
x−a =
g(a)f(x)−f(a)
x−a −f(a)g(x)−g(a) x−a g(x)g(a) . Lettingx→a, using the continuity ofgat a, yields (d).
The preceding theorem, together with 4.1.2 and 4.1.3, show that polyno- mials, rational functions, and trigonometric functions are differentiable. (See Exercise 2.) The following important result will yield additional examples.
4.1.7 Chain Rule. Let g be differentiable ataand let f be differentiable at g(a). Thenf◦g is differentiable ataand(f◦g)0(a) =f0(g(a))g0(a).
Proof. Setb:=g(a). By 4.1.4, there exists a functionη, defined in a neighbor- hoodN(b) ofband continuous atb withη(b) =f0(b), such that
f(y) =f(b) +η(y)(y−b), y∈ N(b). (4.2) Sincegis continuous ata, we may choose a neighborhoodN(a) ofasuch that g(N(a))⊆ N(b). Thenf◦gis defined on N(a), and by (4.2)
f(g(x))−f(g(a))
x−a =η(g(x))g(x)−g(a)
x−a , x∈ N(a)\ {a}.
Lettingx→aproduces the desired result.
Differentiation onR 77 The formula (f◦g)0(x) =f0(g(x))g0(x) is sometimes easier to apply when written in Leibniz notation as
dy dx = dy
du du
dx, wherey=f(u) andu=g(x). 4.1.8 Example. The power rule
Dxr=rxr−1, r∈Q,
follows from 4.1.2 and the chain rule: Letr=m/n,m, n∈N, and setu=x1/n andy=um. Theny=xr and
dy dx = dy
du du
dx =mum−11
nx1/n−1=m
n xm/n−1=rxr−1.
The case r <0 may be verified using the quotient rule. ♦ Higher order derivatives ofy=f(x) are defined inductively by
f00=D2f = d2y dx2 := d
dx dy dx, ...
f(n)=Dnf = dnf dxn := d
dx dn−1f dxn−1. By convention, we setf(0) =D0f :=f.
Exercises
1. Use the limit definition to find the derivative of (a) x2+x+ 1. (b)S √
2x+ 1. (c) 1
x2+ 1. (d)S 1
√3x+ 2. 2. Use the techniques of 4.1.3 to find the derivative of cosx. Use rules of
differentiation to obtain the derivatives of tanx, cotx, secx, and cscx. 3. Use rules of differentiation to findf0 for each of the functionsf:
(a)S √3
5x+ 7√5
3x+ 2. (b) 2x+ 5 7x+ 2
2/3
. (c)S sinx2−1 x2+ 1
. (d) sin2x−1
sin2x+ 1. (e) tancos(1/x)
. (f) q ax+√
bx+c.
4. Assuming thaty is a differentiable function ofxthat satisfies the given equation, use the rules of differentiation to find dy
dx:
(a)x3+y3−xy= 1. (b)S sin(xy2) +x2= 1. (c) tan(x+y) +y2=x.
78 A Course in Real Analysis 5. Letf(x) =xn|x|,n∈N. Findf(n−1)andf(n).
6. Letf(x) =xmbxc,m∈N. Findf`0(n) andfr0(n),n∈Z.
7.S Find all values ofa,b, such thatf0 exists onR, where f(x) =
(ax2+bx+a/x ifx >1,
x3 ifx≤1.
8. Find all values ofa,b, andcsuch thatf0is continuous on (0,+∞), where f(x) =
(ax2+bx ifx >1, c√
x if 0< x≤1. 9. Let
f(x) =
(ax2+bx+c ifx >1,
x3 ifx≤1.
Find all values ofa,b, and csuch that
(a)f is continuous onR. (b)f is differentiable onR. (c)f0 is continuous onR. (d)f00exists onR.
10. Find all values ofcsuch thatf0(c) exists, where f(x) =
(ax−4 ifx > c, 9x2 ifx≤c.
Isf0 continuous at these values?
11. Let f be differentiable at a. Use the limit definition of derivative to calculate
(a) lim
h→0
f(a+ 5 sinh)−f(a+ 2 sinh)
h . (b)S lim
h→0
f(a+h2)−f(a−h)
h .
12. Letg be differentiable on an open intervalI and letf(x) =g(x)d(x), whered(x) is the Dirichlet function (3.1.7). Let abe a zero ofg. Prove that f0(a) exists iffais a zero ofg0.
13. Letf be differentiable atcand let{an}and{bn}be sequences such that an< c < bn andan, bn→c. Prove that
n→∞lim
f(bn)−f(an)
bn−an =f0(c).
14.SLet f be differentiable and increasing on (a, b). Prove thatf0(x)≥0 for allx∈(a, b).
Differentiation onR 79 15. Letf be differentiable ataand nonnegative in a neighborhood ofawith
f(a) = 0. Prove thatf0(a) = 0.
16.S Prove Leibniz’s rule: Iff andg arentimes differentiable, then Dn(f g) =
n
X
k=0
n k
(Dkf) (Dn−kg).
17. Prove that iff has right-hand and left-hand derivatives ata(not neces- sarily equal), thenf is continuous ata.
18. Assuming thatf,g, andhhave the necessary differentiability, find general formulas for
(a) D
f◦(gh). (b)D
f◦(g/h). (c)S D2
f◦g. (d)D
f ◦g◦h. 19. Find a formula for thenth derivative of
(a)S 1/x. (b) 1/√
x. (c)xex. (d)xe−x. 20. Find all values ofp∈Rfor which the function
f(x) =
(|x|psin(1/x) if x6= 0,
0 otherwise
is (a) continuous, (b) differentiable, (c) continuously differentiable onR.
21.S Definef(0) = 0 andf(x) =xmsinxn,x6= 0, wherem∈Z,n∈N. For what values ofmandndoesf0(0) exist? For which of these values isf0 continuous onR?
22. A functionf defined on a symmetric neighborhood (−a, a) of 0 is said to beodd iff(−x) =−f(x) andeven iff(−x) =f(x).
(a) Prove that any function h : (−a, a) → R is the sum of an even functionf and an odd functiong.
(b) Prove that iff is differentiable and odd (even), then f0 is even (odd).
(c) Is the converse true? That is, iff0 is even (odd), is f odd (even)?
23.S Letfj,gj, andhj be differentiable,j= 1,2,3. Prove that
f1 f2 g1 g2
0
=
f10 f20 g1 g2 +
f1 f2 g01 g02 and
f1 f2 f3 g1 g2 g3 h1 h2 g3
0
=
f10 f20 f30 g1 g2 g3 h1 h2 h3
+
f1 f2 f3 g01 g20 g30 h1 h2 h3
+
f1 f2 f3 g1 g2 g3 h01 h02 h03 .
80 A Course in Real Analysis