1549901369-Elements_of_Real_Analysis__Denlinger_

6.2 Rules for Differentiation 311

By the sequential criterion for derivatives, (f-^1 )' (J(x 0 )) = -(

1

)' •

f' Xo

Example 6.2.5 Consider the function f(x) = 3x + 5.

The inverse function is f-^1 (x) = x;

5

. The derivatives off and f-^1 are
f'(x) = 3 and (f-^1 )' (x) = ~· D

Example 6.2.6 Consider the function f(x) = x^2 on the interval [O, +oo).

This function is 1-1 on [O, +oo) and the inverse function for this inter-

val is f-^1 (x) = y'x. Theorem 6.2.4 says that (f-^1 )' (J(x)) = f'~x); that is,

(f-^1 )' (x^2 ) =_!_or (f-^1 )' (y) = -

1

-. This is consistent with the formula you
2x 2,jfj
d 1
remember from elementary calculus: -d y'x = r,;· (See Example 6.1.5.) D
x 2vx

DERIVATIVES OF RATIONAL POWER FUNCTIONS

Having proved Theorem 6.2.4, we are now able to extend the power rule
d~xn = nxn-l to rational exponents. Up to this point we have proved this rule
only for integers n. In fact, until Section 5.5 (Exercise 5.5.15) we did not even
have a definition of xr for general r E Q. Our proof will be in two steps.

Theorem 6.2. 7 Let n E N, n =f-0. The function x:!. is differentiable every-

where on its domain, except at 0, and d~ x:!. = ~ x:!.-^1 if x =f-0.

Proof. Let n E N, n =f-0. Everywhere on its domain, the function x:!. is
1-1 and continuous, and is the inverse of the function xn (see E xercise 5.5.15).
Thus, letting y = f(x) = xn (restricting the domain to [O, oo) if n is even) we
have x = f-^1 (y) = y:!. and by Theorem 6.2.4,

By substituting y = xn (x = y:!.) into this equation, we have

u-1)' (y) = nx:-1

1

In summary, if g(y) = y:!., then g'(y) = ~ y:!.-^1. •