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Placement Preparation
**Specialization:
**Quantitative Aptitude**Offline Downloads:
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The Chain Rule
The chain rule is the last of the derivative formulas. It is most easily written using variables and
the Leibniz notation:
CHAIN RULE
If C depends on ?, and ? depends on B, then:
.C
.C .?
œ
†
.B
.? .B
The idea is that the two .?'s cancel:
y
.C .?
.C
†
œ
y
.B
.? .B
Though this may seem obvious, remember that .B, .C, and .? are “infinitesimals”, which can't
really be treated like regular numbers. The Leibniz notation makes the chain rule look obvious,
but we are actually on slightly shaky ground when we talk about canceling two infinitesimals.
In any case, the chain rule is true, and it is incredibly useful for taking derivatives of
complicated formulas.
EXAMPLE 1 Find
SOLUTION
.C
if C œ ÈB$ ".
.B
Let ? œ B$ ". Then C depends on ?, and ? depends on B:
C œ È?
and
? œ B$ "
We take the derivatives of both of these formulas:
.C
"
œ
.?
#È ?
and
.?
œ $B#
.B
and then multiply the results together:
.C
.C .?
"
œ
†
œ
† $B#
.B
.? .B
#È ?
Finally, we can plug in the formula for ? to express .CÎ.B in terms of B:
.C
"
œ
† $B#
$
È
.B
# B "
è

.C
"
if C œ ŒB# .
.B
B
&
EXAMPLE 2 Find
SOLUTION
Let ? œ B#
"
. Then C depends on ?, and ? depends on B:
B
"
C œ ?&
and
? œ B#
B
Therefore:
.C
"
"
"
œ &?% Œ#B # œ &ŒB# Œ#B #
.B
B
B
B
í ï
.C
.?
.?
.B
%
è
Aside from the power rule, the chain rule is the most important of the derivative rules, and we
will be using the chain rule hundreds of times this semester. Because of this, it is important that
you get used to the pattern of the chain rule, so that you can apply it in a single step. In any case
where:
C œ asomethingb&
the derivative will always be:
.C
œ &asomethingb% † athe derivative of the somethingb
.B
The “something” is what we have been calling ?, but it's fastest not to even think of it as a
variable. If you are given a formula like:
C œ ˆ)B$ $B #‰
&
you should just be able to write down the derivative:
.C
%
œ &ˆ)B$ $B #‰ † ˆ#%B# $‰
.B
Similarly, the derivative of:
C œ Èsomething
is always:
.C
"
œ
† athe derivative of the somethingb
È
.B
# something
For example, if you are given the formula:
C œ È B& B

you should immediately be able to write the derivative:
.C
"
œ
† ˆ&B% "‰
&
È
.B
# B B
Combining Rules
For complicated formulas, you sometimes need to combine different rules together. Here are a
few examples:
EXAMPLE 3 Find
SOLUTION
.C
&
if C œ ˆB# $‰ ÈB.
.B
Here C is a product of two other quantities:
C œ @A,
where @ œ ˆB# $‰ and A œ ÈB
&
We should therefore use the product rule. However, the chain rule will also come into play when
we take the derivative of @:
.C
"
&
%
œ ˆB# $‰ †
ÈB † &ˆB# $‰ † #B
.B
#È B
í ðóóóñóóóò
ðñò í
.A A
.@
@
.B
.B
EXAMPLE 4 Find
è
.C
B$
if C œ
.
È " B#
.B
This requires the quotient rule, but the chain rule will come into play when we take
the derivative of È" B# :
SOLUTION
.C
œ
.B
ŠÈ" B# ‹a$B# b aB$ b
" B#
"
† #B
È
# " B#
è

Related Rates
The chain rule can also be used to find a relationship between two unknown derivatives. For
example, suppose that:
C œ ?$
.C
? This is similar to a normal chain rule problem, except that we don't have a formula
.B
for ? in terms of B. The best we can do is write:
What is
.C
.?
œ $?# †
.B
.B
í í
.C .?
.? .B
This follows the same basic pattern as any other chain rule: the derivative of asomethingb$ is
$asomethingb# multiplied by the derivative of the something.
EXAMPLE 5 Find
SOLUTION
.C
if C œ È?.
.B
Using the chain rule:
.C
"
.?
œ
†
.B
#È? .B
í í
.C
.?
.?
.B
è
If the variable B is time, this method can be used to find the relationship between two different
rates of change:
EXAMPLE 6 The side length P and area E of a square are related by the equation:
E œ P#
What is the relationship between
SOLUTION
.E
.P
and
?
.>
.>
Using the chain rule:
.E
.P
œ #P †
.>
.>
í í
.E .P
.P .>
è

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