Calculus: Analytic Geometry and Calculus, with Vectors

9.2 Derivatives and integrals of exponentialsand logarithms 491

But (H''Q)XP = (H),P)xQ and (HxQ)P = (H>P)Q, and it follows that each member of
(5) is HxP. Thus (5) is true and this proves (4). Just as the rational numberr
defined by r = -4$ is representable in the forms

(^63) (7)(-9) -9

r -28 - (7)(4)

_

-4-'

so also each rational number r given in the form r = P/Q, where P and Q are
integers, is uniquely representable in the form p/q, where P = Ap, Q= Xq, and
A, p, and q are integers for which q > 0 and the integers IpI andq are relatively
prime, that is, have no common positive integer factor different from 1. This
fact and (4) show that when P and Q are integers for which Q 0, we can define
aP/Q by the first of the formulas

(6) aP/Q = (aP)1/Q = (al/Q)P
a
with assurance that the whole formula is correct and that the formula remains
correct when we multiply or divide both P and Q by the same integer X provided
A 0 when we divide. This completes the definition of ax whenx is rational.
To prove that the laws (2) hold when x and y are rational, we can letx = p/n
and y = q/n, where p, q, n are integers and n > 0, to obtain

axay = (aP/n)(a4/n) = (a11n)P(al/n)Q = (alln)P+Q = a(P+4)ln = ax+y
and

so

((ax)y)n2 = ((\01n14)1/nln2 = (((aP)1/n)Q)n = (((a")Q)lln)n = aP4

(ax),l = (aPs)1/n2 = aPQln2 = axy.

9.2 Derivatives and integrals of exponentials and logarithms

Let a > 1. Looking forward to a derivation of a formula for the deriva-

tive of the function f for which f(x) = ax, we use h instead of Ax, write

(9.21)

and observe that

ax+h - ax= (all - 1) az,

d

=

ax+h -ax ah-1

(9.22)

dx

ax ho

h

h-0 h

J ax

provided the limits exist. If we succeed in proving that, for some con-

stant A which may depend upon a, the first of the formulas

(9.221) lim ah

h

1

ah

= A,

hlli m h

1= A

is valid, then the second will also be valid because

ah -

1

h -

1 1

h h -

lim = lim a = lim lim a '' a 1

a

=h_

h-.0- h h-+0+ -h h-O+ h h

and we will be able to conclude that

=A

(9.222)

TX

ax = Aax.