3.3 Product Rule
Wed Oct 14
1)
2)
3)
Do Now
Evaluate each
d
dx
x
d
dx
x ×
d
dx
4x - x
7
2
5
d
dx
x
5
1/ 2
HW Review p.139 #15-41 4953 61 71 75
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•
•
•
•
•
•
•
•
•
15) y’ = 4x^3, y = 32x - 48
17) y’ = 5 - 16x^-1/2, y = -3x - 32
19) a) 12e^x
b) 25 - 8e^t c) e^(t-3)
21) f’(x) = 6x^2 - 6x
23) f’(x) = 20/3 x^2/3 + 6x^-3
25) g’(z) = -5/2 z^-19/14 - 5z^-6
27) f’(s) = 1/4 s^-3/4 + 1/3 s^-2/3
29) g’(x) = 0
31) h’(t) = 5e^(t-3)
33) p’(s) = 32s - 24
HW Review cont’d
•
•
•
•
•
•
•
•
35) g’(x) = -6x^-5/2
37) 1
39) -60
41) 1 - e^4
49) a) 3ct^2
b) 5 + 8cz
c) 27c^2 y^2
51) x = 1/2
53) a = 2 and b = -3
61) decreasing; y = -0.63216(m - 33) + 83.445
y = -0.25606(m - 68) + 69.647
• 71) c = 1
• 75) c = +/- 1
3.1 3.2 Quiz
3 problems
15 pts?
1)3.1 hw #51 53 55
2)Find derivative using limit def f(x+h) – f(x) / h
3.2 #1-6
3)Matching graphs
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: f(x) = x e^x
Ex 1
2
f
(x)
=
x
(9x + 2)
• Use the product rule to find
Ex 2
• Find f’(x) if
-1
f (x) = (2 + x )(x
3/2
+1)
Ex 3
• Find the derivative of the function
y = (x 4 - 3x 2 + 2x)(x 3 - 2x + 3)
Closure
• Hand in:
• Compute the derivative of
(3x +14)(2x + 3x - 2)
2
3
Using the product rule
HW: none
3.3 Product Rule
Thurs Oct 15
Do Now
Evaluate each
3
2
d
1) dx x (2x - 4 x + 2)
2) d
1/ 2
dx
(3x - x )(x + 3)
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: f(x) = xe^x
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)]
• This is especially useful when we
cannot simplify the fraction.
2
Ex 1
x
e
f (x) = x
• Find the derivative of
e +x
Ex 2
• Find the derivative of
x2 - 2
f (x) = 2
x +1
Ex 3
3x 2 + x - 2
f (x) =
4 x 3 +1
• Find the tangent line to the graph of f(x)
at x = 1
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 + 22
x
=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 p.146
Product/Quotient Rule
Practice
• #1-2 Find the derivative
• 1) f (x) = x (1- x 4 )
• 2)
x+4
f (x) = 2
x + x +1
• 3) Find the equation of the tangent line to
x2
at x = 9
f (x) =
x+x
Closure
• Find the derivative of each:
f (x) = (3x - 5)(2x - 3)
2
x+4
g(x) = 2
x + x +1
• HW: p.147 #1-45 odds 49 51 57 59
3.3 HW Review
Fri Oct 16
• Do Now
• Find the derivative of each
• 1) f (x) = x 2 (3 + x -1 )
x + 2x + 3x
• 2) g(x) =
x
3
2
-1
HW Review
• P.147 #1-45 odds 49 51 57 59
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?
• Redo any HW problems not understood