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Transcript 
Section 2.4: The Chain Rule
Goals for this Section:
Find the derivative of a composite function using the
Chain Rule.
Find the derivative of a function using the General Power
Rule.
Simplify the derivative of a function using algebra.
Find the derivative of a trigonometric functions using the
Chain Rule.
1
Section 2.4: The Chain Rule
Theorem 2.10: The Chain Rule
If y = f (u) is a differentiable function of u and
u = g (x) is a differentiable function of x, then y =
f (g (x)) is a differentiable function of x and
dy
dx
=
dy
du
•
du
dx
or, equivalently,
d [ f (g (x)) ] =
f ' (g (x))•g ' (x)
dx
d
dx
f (u)
=
f ' (u)•u'
2
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) = (2x - 5)3
3
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) =
3 - 2x
4
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) =
x ( 3x - 9 )3
5
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) =
(
3x - 1
x2 + 3
(
2
6
Section 2.4: The Chain Rule
Theorem 2.11: The General Power Rule
If y = [ u (x) ]n , where u is a differentiable
function at x and n is a rational number, then
dy
dx
=
n[ u (x)
]n-1
du
dx
or, equivalently,
d
n] =
[
u
dx
n • un-1 u'
7
Section 2.4: The Chain Rule
Trigonometric Functions and the Chain Rule:
d
dx
[ sin u ]
(cos u)
•
u'
d
dx
[ cos u ] = -(sin u)
•
u'
d
dx
[ tan u ] = (sec2 u)
•
u'
d
dx
[ cot u ] = -(csc2 u)
d
dx
[ sec u ] = (sec u • tan u )
d
dx
[ csc u ] = -(csc u • cot u )
=
•
u'
•
u'
•
u'
8
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) = sin 2x
9
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) = cos x2
10
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) = (cos x)2
11
Section 2.4: The Chain Rule
Example: Find the derivative of the function.
f (x) = sin3 4x
12
Section 2.4: The Chain Rule
Example: Evaluate the derivative of the function at
the given point.
Function
f (x) =
5
Point
3x3 + 4x
(2,2)
13
Section 2.4: The Chain Rule
Example: Find an equation of the tangent line to the
graph of f at the given point.
Function
f (x) = ⅓x
Point
x2 + 5
(2,2)
14
15
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