Calculus: Analytic Geometry and Calculus, with Vectors

3.6 The chain rule and differentiation of elementary functions 167

and all others obtainable by making "finite combinations" of basic
elementary functions together with addition, subtraction, multiplication,
and division. This class contains very many important functions. It is
therefore important to know that we can work out a formula for the
derivative of any given elementary function when we know (i) Theorem
3.56, (ii) 15 basic formulas for derivatives of basic elementary functions,
(iii) the chain rule, and, in addition, we possess (iv) a technique which
enables us to apply these things.
Of the 15 basic formulas, the most important 5 were listed at the
beginning of this section and are relisted in the first column of the follow-
ing little table.

(3.671) d X, = nxi-1

d

un =un-1 du

YX_

(3.672) dx sin x = cos x d sin u = cos udu

(3.673) - cosx = - sinx dxcosu= -sinudx

(3.674) dx ex

=

ex

d

d

eu = eu dx

(3.675) dx log x =

(^1) d
log u =udu
If we know the first formula on the left, we can set y = u" and use the
chain formula (3.66) to obtain the chain formula
dun =dy=dy du= nun-1
du
dx dx du dx dx
written opposite it. If we know the second formula on the left, we can
set y = sin u and use (3.66) again to obtain the chain formula
d dy__dy du du
dx sin u =
dx du dx =(cos u)dx
written opposite it. The same procedure shows that each basic formula
has a chain extension. Of the ten basic formulas not listed above, four
(which appear in Problem 11 of Section 3.5 and have probably been for-
gotten) give derivatives of the last four trigonometric functions, and the
remaining six give derivatives of the inverse trigonometric functions.
Proofs of all of the formulas will appear later. Except for three formulas
that are rarely used, the formulas are listed on the page opposite the
back cover of this book.
Our fund of information about logarithms is quite meager, but we can
slowly add to it. We begin with the idea that log x exists (as a real

number) only when x > 0. In case x < 0, log x does not exist but

IxI > 0 and log IxI does exist. When x < 0, we can use the chain formula