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CHAPTER 3
DIFFERENTIATION RULES
Observe that
is a composite function. In fact, if we let
and let
, then we can write
, that is,
. We know
how to differentiate both
and
, so it would be useful to have a rule that tells us how to
fi
nd the derivative of
in terms of the derivatives of
and .
It turns out that the derivative of the composite function
is the product of the deriv-
atives of
and
. This fact is one of the most important of the differentiation rules and is
called the
Chain Rule.
It seems plausible if we interpret derivatives as rates of change.
Regard
as the rate of change of
with respect to ,
as the rate of change of
with respect to , and
as the rate of change of
with respect to . If
changes
twice as fast as
and
changes three times as fast as
, then it seems reasonable that
changes six times as fast as
, and so we expect that
The Chain Rule
If
f
and
t
are both differentiable and
is the composite func-
tion de
fi
ned by
, then
is differentiable and
is given by the
product

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