James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.4 The Chain Rule 199
The only flaw in this reasoning is that in (1) it might happen that Du − 0 (even when
Dx ± 0) and, of course, we can’t divide by 0. Nonetheless, this reasoning does at least
suggest that the Chain Rule is true. A full proof of the Chain Rule is given at the end of
this section.
■
The Chain Rule can be written either in the prime notation
2 s f 8 td9sxd − f 9stsxdd ? t9sxd
or, if y − f sud and u − tsxd, in Leibniz notation:
3
dy
dx − dy
du
du
dx
Equation 3 is easy to remember because if dyydu and duydx were quotients, then we
could cancel du. Remember, however, that du has not been defined and duydx should not
be thought of as an actual quotient.
ExamplE 1 Find F9sxd if Fsxd − sx 2 1 1.
SOLUTION 1 (using Equation 2): At the beginning of this section we expressed F as
Fsxd − s f 8 tdsxd − f stsxdd where f sud − su and tsxd − x 2 1 1. Since
f 9sud − 1 2 u21y2 − 1
2su
and t9sxd − 2x
we have
F9sxd − f 9stsxdd ? t9sxd
−
1
2sx 2 1 1 ? 2x −
x
sx 2 1 1
SOLUTION 2 (using Equation 3): If we let u − x 2 1 1 and y − su , then
F9sxd − dy
du
du
dx − 1 s2xd −
2su
1
2sx 2 1 1 s2xd −
x
sx 2 1 1
■
When using Formula 3 we should bear in mind that dyydx refers to the derivative of
y when y is considered as a function of x (called the derivative of y with respect to x),
whereas dyydu refers to the derivative of y when considered as a function of u (the
derivative of y with respect to u). For instance, in Example 1, y can be considered as a
function of x (y − sx 2 1 1) and also as a function of u (y − su ). Note that
dy
dx − F9sxd −
x
sx 2 1 1
whereas
dy
du − f 9sud − 1
2su
NOTE In using the Chain Rule we work from the outside to the inside. Formula 2
says that we differentiate the outer function f [at the inner function tsxd] and then we
multiply by the derivative of the inner function.
d
dx
f stsxdd − f 9 stsxdd ? t9sxd
outer
function
evaluated
at inner
function
derivative
of outer
function
evaluated
at inner
function
derivative
of inner
function
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