3.7 Chain Rule
Fri Nov 13
Do Now
Find the derivative of each
2
(use product rule)
1)
sin x
2)
(x +1)
2
2
HW Review p.167 #5-31 43
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5) -sin2 x + cos2 x
23) eq (5sinq + 5cosq - 4tanq - 4sec 2 q )
7) 2sin x cos x
25) y = 1
2
2sint
sec
t tant + sec t 27) y = x + 3
9)
p
2
2
y
(1+
3)
=
(13)(x
3)
11)(tan q + sec q )secq 29)
4
-1
3
-2
(2x
4x
)sec
x
tan
x
+
sec
x(8x
+
4x
)
13)
q secq tanq - secq
y = x +1
2
15)
31)
q
4 cos y - 3
17)
43) -sinx, -cosx, sinx, cosx,
2
sin
y
2
2sec
x
19)
-sinx
2
(1 - tan x)
21) ex (cos x + sin x)
f^8(x) = cosx f^37(x) = -sinx
Chain Rule
• Thm- If f and g are differentiable,
d
dx
[ f (g(x))] = f ¢(g(x))× g¢(x)
• In words, the theorem says “the derivative of
the outside times the derivative of the inside”
Examples of “Inside”
• A whole expression being raised to a
power
3
5
(5x + 2)
• An expression inside of a trig function
sin3x
2
• An expression inside of an exponent
4e
3x +2
-1
Chain Rule cont’d
• Another way to think of the chain rule is
to use U-substitution
• We let u = whatever is inside the
parenthesis, then differentiate.
Ex 7.1
• Differentiate
y = (x + x -1)
3
5
Ex 7.2
• Find
d
dt
(e
-t / 6
)
Ex 7.3
• Suppose the position of a weight
hanging from a spring is given by
-t / 6
u(t) = e
cos4t
• Find the velocity of the weight at any
time t
Chain Rule and Trig
• Compute the derivatives of
f (x) = cos x
3
g(x) = cos x h(x) = cos3x
3
Closure
• Journal Entry: When do we use the
chain rule? What is the chain rule?
• HW: p.175 #11-21, 29-41 odds
3.7 Multiple Chain Rule
Mon Nov 16
• Do Now
• Use chain rule to differentiate:
• 1) (3x + 2) 2
• 2)
sin(3x + x)
2
HW Review p.175 #11-21 29-41
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11) 12x 3 (x 4 + 5) 2
-1/ 2
7
13) 2 (7x - 3)
2
-3
-2(x
+
9x)
(2x + 9)
15)
17) -4cos3 q sinq
19) 9(2cosq + 5sinq )8 (5cosq - 2sinq )
21) e x -12
29) 2x cos(x 2 )
31) t(t 2 + 9)-1/ 2
33) 23 (x 4 - x 3 -1)-1/ 3 (4x 3 - 3x 2 )
3
8(1+
x)
35)
(1 - x) 5
HW Review
• 37) -sec(1/ x)tan(1/ x)
x2
2
(1sin
q
)sec
(q + cosq )
• 39)
• 41)
-18te
2-9t 2
Multiple Chain Rules
• The more complicated a function gets,
the more we will use the chain rule to
differentiate
• Set up the derivative before
differentiating!
Ex
• Differentiate
f (x) = (sin x +1)
2
5
Ex 7.6
tan2x 2
• Find the derivative of f (x) = (2x + 1) 2
Ex 7.7
• Find the derivative of
f (x) = [cos(e
x -sec x
2
4
)]
Practice:
• Bookwork: p.175 #43-69 odds
Closure
• Hand in: Find the derivative of:
(2cos x + 3)
2
2/3
• HW: p.175 #43-67 odds skip 57 59
3-7 Chain Rule
Practice
Tues
Nov
17
Do Now
•
• Use chain rule to find each derivative
• 1)
3x 2
y = 3e
• 2)
y = cossin x
2
HW Review p.175 #43-69
2
2
(2x
+
4)sec
(x
+ 4x)
• 43)
• 45) 3x sin(1- 3x) + cos(1- 3x)
-1/ 2
2(4t
+
9)
• 47)
2
3
-5
4(sin
x
3x
)(x
+
cos
x)
• 49)
-1/ 2
(2sin2x)
cos2x
• 51)
(cos6x + sin(x 2 )-1/ 2 (x cos x 2 - 3sin6x)
• 53)
• 55) 3(x 2 sec2 (x 3 ) + sec2 x tan2 x)
4
6
5
2
5
-35x
cot
(x
)csc
(x
)
• 61)
63 65 67
3
4
4
2
4
5
4
8
-180x
cot
(x
+1)csc
(x
+1)(1+
cot
(x
+1))
• 63)
3x
-2x 3 3x
-2x
24(2e
+
3e
)
(e
e
)
• 65)
(x 2 +2x+3)2
• 67) 4(x +1)(x + 2x + 3)e
2
Practice
• (blue book) Worksheet p.214 #7-23, 31,
32, 35 skip 13, 21 22
Closure
• Hand in: Find the derivative of:
y = (sin(e 2x ))1/ 2
• HW: worksheet p.214 #7-26, 31, 32, 35
3.7 Chain Rule Practice Day 2
Wed Nov 18
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Do Now
Use the chain rule to find dy/dx
2
-1/ 3
1) y = (x + 7x + 2)
2)
y =e
cosx
HW Review worksheet 214
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7) 37(x^3+2x)^36 (3x^2+2)
8) 6(3x^2+2x-1)^5 (6x+2)
9) -2(x^3-7x^-1)^-3 (3x^2 +7x^-2)
10) -9(x^5-x+1)^-10 (5x^4-1)
11) -12 (3x^2-2x+1)^-4 (6x-2)
12) ½ (x^3-2x+5)^-1/2 (3x^2-2)
14) ¼ x^-3/4
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15) cos(1/x^2) (-2x^-3)
16) sec^2(x^1/2) (1/2 x^-1/2)
17) 20(cosx)^4 (-sinx)
18) 4 + 20(sinx)^3 (cosx)
19) 2(cos3x^1/2) (-sin3x^1/2) (3/2x^-1/2)
20) 4(tanx^3)^3 (sec^2x^3) (3x^2)
23) ½(cos5x)^-1/2 (-sin5x) (5)
31) –sincosx (-sinx)
32) costan3x (sec^2 3x) (3)
35) product rule
More Practice
• (green book) worksheet p.219 #9-28 32
33, skip 23 24
Closure
• Journal Entry: If you had to describe
how to use the chain rule to your friend,
how would you?
• HW: Finish worksheet