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The Chain Rule
1. Learning Objectives for Section 11.4 The Chain Rule
The student will be able to form thecomposition of two functions.
The student will be able to apply the
general power rule.
The student will be able to apply the
chain rule.
Barnett/Ziegler/Byleen Business Calculus 11e
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2. Composite Functions
Definition: A function m is a composite of functions f and g ifm(x) = f [g(x)]
The domain of m is the set of all numbers x such that x is in the
domain of g and g(x) is in the domain of f.
Barnett/Ziegler/Byleen Business Calculus 11e
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3. General Power Rule
We have already made extensive use of the power rule:d n
x nx n 1
dx
Now we want to generalize this rule so that we can
differentiate composite functions of the form [u(x)]n,
where u(x) is a differentiable function. Is the power rule
still valid if we replace x with a function u(x)?
Barnett/Ziegler/Byleen Business Calculus 11e
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4. Example
Let u(x) = 2x2 and f (x) = [u(x)]3 = 8x6. Which of thefollowing is f ’(x)?
(a) 3[u(x)]2
(b) 3[u’(x)]2
Barnett/Ziegler/Byleen Business Calculus 11e
(c) 3[u(x)]2 u’(x)
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5. Example
Let u(x) = 2x2 and f (x) = [u(x)]3 = 8x6. Which of thefollowing is f ’(x)?
(a) 3[u(x)]2
(b) 3[u’(x)]2
(c) 3[u(x)]2 u’(x)
We know that f ’(x) = 48x5.
(a) 3[u(x)]2 = 3(2x2)2 = 3(4x4) = 12 x4. This is not correct.
(b) 3[u’(x)]2 = 3(4x)2 = 3(16x2) = 48x2. This is not correct.
(c) 3[u(x)]2 u’(x) = 3[2x2]2(4x) = 3(4x4)(4x) = 48x5. This is the
correct choice.
Barnett/Ziegler/Byleen Business Calculus 11e
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6. Generalized Power Rule
What we have seen is an example of the generalized powerrule: If u is a function of x, then
d n
n 1 du
u nu
dx
dx
For example,
d 2
( x 3 x 5)3 3( x 2 3 x 5) 2 (2 x 3)
dx
du
2
Here u is x 3 x 5 and
2x 3
dx
Barnett/Ziegler/Byleen Business Calculus 11e
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7. Chain Rule
We have used the generalized power rule to find derivativesof composite functions of the form f (g(x)) where f (u) = un
is a power function. But what if f is not a power function?
It is a more general rule, the chain rule, that enables us to
compute the derivatives of many composite functions of the
form f(g(x)).
Chain Rule: If y = f (u) and u = g(x) define the
composite function y = f (u) = f [g(x)], then
dy dy du
dy
du
, provided
and
exist .
dx du dx
du
dx
Barnett/Ziegler/Byleen Business Calculus 11e
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8. Generalized Derivative Rules
1.d
n
n 1
f x n f x f ' ( x)
dx
If y = u n , then
y’ = nu n - 1 du/dx
2.
d
1
ln [ f ( x)]
f ' ( x)
dx
f ( x)
If y = ln u, then
y’ = 1/u du/dx
3.
d f ( x)
e
e f ( x ) f ' ( x)
dx
If y = e u, then
y ’ = e u du/dx
Barnett/Ziegler/Byleen Business Calculus 11e
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9. Examples for the Power Rule
Chain rule terms are marked:y x , y' 5x
5
4
y (2 x) , y ' 5(2 x) (2) 160 x
5
4
4
y (2 x 3 ) 5 , y ' 5(2 x 3 ) 4 (6 x 2 ) 480 x14
y (2 x 1) 5 , y ' 5(2 x 1) 4 (2) 10(2 x 1) 4
y (e x ) 5 , y ' 5(e x ) 4 (e x ) 5e 5 x
y (ln x) 5 , y ' 5(ln x) 4 (1 / x)
Barnett/Ziegler/Byleen Business Calculus 11e
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10. Examples for Exponential Derivatives
d uu du
e e
dx
dx
y e 3 x , y ' e 3 x (3) 3e 3 x
y e
3 x 1
y e
4 x 2 3 x 5
y e
ln x
, y' e
3 x 1
, y' e
x, y ' e
Barnett/Ziegler/Byleen Business Calculus 11e
(3) 3e
4 x 2 3 x 5
ln x
3 x 1
(8 x 3)
1
x
( ) 1
x
x
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11. Examples for Logarithmic Derivatives
d1 du
ln u
dx
u dx
1
1
y ln( 4 x), y '
4
4x
x
1
4
y ln( 4 x 1), y '
4
4x 1
4x 1
1
2
2
y ln( x ), y ' 2 (2 x)
x
x
1
2x 2
2
y ln( x 2 x 4), y ' 2
( 2 x 2) 2
x 2x 4
x 2x 4
Barnett/Ziegler/Byleen Business Calculus 11e
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