Calculus: Analytic Geometry and Calculus, with Vectors

3.6 The chain rule and differentiation of elementary functions 165

formulas

(3.61) dx xn = xn-1, x sin x =cos x, x cos x = - sin x

(3.62) dxd d

ex = ex, dxlog x x

only the first of which has been partially proved.t In the last two of
these formulas, the base is e, the base of natural exponentials and loga-
rithms, which appears in (3.272) and which will appear later. One of our
tasks is to learn a procedure by which we can obtain a correct formula
for dy/dx when y = sin is and u is a differentiable function of x which is
not necessarily x itself. The answer is

621 dy du

( 3. ) dx =cos is

dx

To see why this is so, and to see how many similar formulas can be

obtained, we consider the general situation in which y is a function ofis
and is is a function of x, say y = f(u) and is = g(x). Then y is linked to x
through the links of a short chain; x determines is and is determines y, so y
is a function of x. While the operation may seem somewhat ponderous
when

(3.622) y = ¢(x) = f(g(x)) = sin g(x) = sin is = sin 2x,

we can let q5(x) = f(g(x)) and sketch the schematic Figure 3.63 which
catches the functions g, f, and 0, re-
spectively, in the act of transforming
(or mapping or carrying) x into is, is
into y, and x into y. The function 0
for which 4(x) = f(g(x)) is sometimes
called a composite function.
The following theorem is the chain

Figure 3.63

rule, which sets forth conditions under which y has a derivative with
respect to x that can be calculated from the chain formula

(3.64)

dy _ dy du

dx du T

The result is given in terms of the "d" notation of Leibniz and the
"prime" notation of Newton, so that we can, in applications, choose the
one that seems to be most convenient or informative.

t These formulas will be proved in Chapters 8 and 9. A contention that we can and
should learn and use these formulas before they are proved is pedagogically sound. It is as
practical as the contention that embryonic electrical engineers should learn that copper
wires conduct electric current, and use this information in various ways, before they study
solid-state physics and learn mechanisms by which electrons travel along conductors and
semiconductors.