1549901369-Elements_of_Real_Analysis__Denlinger_

312 Chapter 6 11 Differentiable Functions

Corollary 6.2. 8 l::/r E Q, the power function f(x) = xr is differentiable every-

where on its domain, and fxxr = rxr-^1.

Proof. Exercise 10. •

LOGARITHM, EXPONENTIAL, AND POWER FUNCTIONS

Students who skipped Section 5.6 should skip the proofs of

the following theorem and its corollaries. These results will

be derived in the more customary way in Chapter 7.

We adopt the conventional notation for the natural logarithm, lnx =loge x.
This function and ex were treated in detail in Section 5 .6.

Theorem 6.2.9 The functions f(x) = lnx and g(x ) = ex are differentiable

everywhere on their domains, and

d 1 d

(a) -lnx=-; (b) -ex=ex.

dx x dx

*Proof. (a) Using the properties of logarithms,

lim ln( x + h) - ln x = lim ~ ln ( x + h)

h-+O h h-+O h X

1

= lim ln ( x + h) T.:

h-+O X

= ln [k~ (1 + ~) *] by Theorem 5.1.14 (b)

= ln e^1 fx by Corollary 5.6.20

1

x

(b) The exponential function ex is the inverse of the logarithm function
lnx. That is, for f(x) = lnx we have f-^1 (x) =ex. Thus, by Theorem 6.2.4
with y = lnx,

d 1 1

• eY = --= - = x = eY •
  dy g dx lnx .!. x ·