3.3 Product Rule
Tues Oct 27
Do Now
Evaluate each
7
d
1) dx x
2) d 4 x 5 - x1/ 2
dx
HW Review p.184 5-23 odds
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•
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9x - x
3
1 -2
-2x
2
2
5) 3x^2 -2
7) 6x
9) 0
2
-1/ 2
11) 9t - t
-2
-3x
-8
13)
15) -5x^(-3/2) -2
17) 3s1/ 2 + s-4 / 3
2 -2 / 3
19)
x
3
21)
23)
3
2
1/ 2
HW Review p.184 5-23 evens
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8x - 6x +1
3
6)
8
4
8) 9x -15x + 8x +1
10) -4 -1/ 2
5
- 8s
12) 2 s
-5
2
-8x
3x
14)
3 -3 / 2
16) 12 - 2x + 2 x
p -1
0.3
18)
20) 4 - 2x -1/ 3
2
22)
3pt
- 2.6t
9x + 6x - 4
Not the Product Rule
• Consider our derivative rules so far. We
do not know the derivative if 2 terms are
multiplied together
• Note: the derivative of a product is NOT
the product of the derivatives:
7
2
5
d
d
d
dx x ¹ ( dx x )( dx x )
7x ¹ (2x)(5x )
6
7x ¹ 10x
6
4
5
Proof
Product Rule
• Thm- Suppose that f(x) and g(x) are
differentiable at x. Then:
d
dx
[ f (x)g(x)] = f ¢(x)g(x) + f (x)g¢(x)
• Ex: y = x e^x
Ex 1
• Use the product rule to find f (x) = x (9x + 2)
2
• Find f’(x) if
Ex 2
-1
3/2
f (x) = (2 + x )(x +1)
Ex 3
• Find the derivative of the function
y = (x - 3x + 2)(x - 2x + 3)
4
2
3
Closure
• Hand in:
• Compute the derivative of
(3x +14)(2x + 3x - 2)
2
3
Using the product rule
HW: p.147 #1-3, 5, 13, 14, 17
3.3 Quotient Rule
Wed Oct 28
• Do Now
• Use the product rule to find the
derivative:
2
3
• 1) (x - 2)(x - x +1)
• 2) (x - 2x +1)(3- 2/ x)
3
2
HW Review: p.147 #1-3 5 13
14 17
• 1) 10x 4 + 3x 2
• 2) 18x 2 - 20x - 9
x
2
e
(x
+ 2x)
• 3)
• 5) 871/64
2
6t
+ 2t - 4
• 13)
• 14) 6x + 1
• 17) 6x 5 + 4x 3 +18x 2 + 5
Quotient Rule
• Thm- Suppose that f and g are
differentiable at x and g(x) not equal to
0, then:
d
dx
[
f (x )
g(x )
]=
f ¢ (x )g(x )- f (x)g ¢(x)
[g(x)]
2
• This is especially useful when we
cannot simplify the fraction.
Ex 1
x
e
• Find the derivative of f (x) = e x + x
Ex 2
• Find the derivative of
x2 - 2
f (x) = 2
x +1
Ex 3
• Find the tangent line to2the graph of f(x)
3x + x - 2
at x = 1
f (x) =
4 x +1
3
Cases where the Product and
Quotient rules are not needed
• Sometimes, it’s easier to simplify and
use the power rule instead of the
product or quotient rule
• Ex: f (x) = x x + 2
x2
=x
3/2
+ 2x
f ¢(x) = x
3
2
-2
1/ 2
- 4x
-3
Applications
• Remember that the derivative may be
used to represent rates such as speed.
• Now that we can differentiate more
complicated functions, we can now
apply these to other types of rates.
• Ex 7
Closure
• Journal Entry: Write about the product
and quotient rules. What are the
formulas? How do we use them? Do we
need to use them for every problem?
• HW: p.147 #7-11, 15, 23, 25
3.2-3.3 Practice
Thurs Oct 29
• Do Now
• Find the derivative of each
• 1) f (x) = (3x1/ 3 -1)(2x + 3)
• 2)
3x + 1
f (x) = 2
2x - x + 1
2
HW Review p.147 #7-11 15
23 25
-2
•
•
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•
(x - 2) 2
7) 2
-x - 8x - 3
8) (x 2 + x +1)2
9) 8/9
10) 27/32
-e x
11)
x 2
• 15) 1
(1+ e )
23) -80
25) -3/4
Derivative Review
• Power Rule
• Product Rule
• Quotient Rule
3.3 Practice
• Start in class #1-18. Complete for HW
Closure
• Find the derivative of:
f (x) = 3e x
2
2x -1
g(x) =
3x + 2
x 1/ 3
• HW: Finish worksheet p.203 #1-18
HW Review: wkst p.203 #1-18
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1) 4x + 1
2) 12x 3 +10x
3
3) 4 x2
4) 3x
2
18x
- 32 x +12
5)
2
5
3
4
6) (-1- 9x )(7 + x ) + (2 - x - 3x )(5x )
2
-3
-4
3
2
-4
-5
(3x
+14x)(2x
+
x
)
+
(x
+
7x
8)(-6x
4x
)
7)
-2
-3
3
-1
-2
2
(-x
2x
)(3x
+
27)
+
(x
+
x
)(9x
)
8)
2
3x
9)
3
10) 4x - 2x
HW Review #11-18
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11) -5/4
12) -1/6
13) 7/16
14) -7/4
15) -29
16) 1/2
17) 0
18) 1