1549901369-Elements_of_Real_Analysis__Denlinger_

6.2 Rules for Differentiation 313

Corollary 6.2.10 Suppose a> 0, and a -=j:. l. Then

d d 1

(a)VxEIR,-d ax=axlna; (b)Vx>O, - logax=-

1

-.

x dx x na

Proof. E xercise 11. •

d
Corollary 6.2.11 If c E JR, then - xc = cxc-^1.
dx
Proof. Exercise 12. •

TRIGONOMETRIC FUNCTIONS

The trigonometric functions will b e defined rigorously in Chapter 7, where
the sine and cosine functions will b e defined using the integral as a foundation.
They will also be defined in Chapter 8 using power series. In either context their
derivatives are also readily obtained. In the meantime, we sh all need to use these
functions as examples. We thus give the following formulas for reference.

Table 6.1

For all real numbers in the domain of the indicated functions,

(a) d~ (sin x) = cos x (b) d~ (cos x) = - sin x

d d

(c) dx (tanx) = sec^2 x (d) dx (secx) = secxtan x

d d

( e) dx (cot) = - csc^2 x ( f) dx ( csc x) = - csc x cot x

The "proof'' of formula (a) usually given in calculus courses is not accept-
able here since at a crucial point it relies on geometric, rather than analytic ,
reasoning. Of course, once formula (a) h as been proved, the remaining five
are easily derived from it using trigonometric identities and the "Algebra of
Derivatives." We shall assume that these functions have their familiar prop-
erties, until we actually prove them in Section 7.7. (See Exercise 16 on page
315.)

EXERCISE SET 6.2

l. Prove Theorem 6.2.2 (c).

2. Without using the chain rule, prove the general power rule for natural
   numbers: if f is differentiable at x 0 , then Vn E N, Jn is differentiable at
   xo, and d~ [f(x)t = n [f(x)t-^1 f'(x). [Use mathematical induction.]