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3.7 Chain RuleTues Nov 4
Do Now
Find the derivative of each
1) (use product rule)
2)
HW Review p.167 #5-31 43
• 5) 23)
• 7) 2sin x cos x 25) y = 1
• 9) 27) y = x + 3
• 11) 29)
• 13)
• 15) 31)
• 17) 43) -sinx, -cosx, sinx, cosx,
• 19) -sinx
• 21) f^8(x) = cosx f^37(x) = -sinx
Chain Rule
• Thm- If f and g are differentiable,
• 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
• An expression inside of a trig function
• An expression inside of an exponent
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
Ex 7.2
• Find
Ex 7.3
• Suppose the position of a weight hanging from a spring is given by
• Find the velocity of the weight at any time t
Chain Rule and Trig
• Compute the derivatives of
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 RuleWed Nov 5
• Do Now
• Use chain rule to differentiate:
• 1)
• 2)
HW Review p.175 #11-21 29-41• 11) • 13)• 15)• 17) • 19)• 21)• 29)• 31)• 33)• 35)
HW Review • 37)
• 39)
• 41)
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
Ex 7.6
• Find the derivative of
Ex 7.7
• Find the derivative of
Practice:
• Bookwork: p.175 #43-69 odds
Closure
• Hand in: Find the derivative of:
• HW: p.175 #43-67 odds skip 57 59
3-7 Chain RulePractice
Mon Nov 10• Do Now
• Use chain rule to find each derivative
• 1)
• 2)
HW Review p.175 #43-69• 43)
• 45)
• 47)
• 49)
• 51)
• 53)
• 55)
• 61)
63 65 67
• 63)
• 65)
• 67)
Practice
• (blue book) Worksheet p.214 #7-23, 31, 32, 35 skip 13, 21 22
Closure
• Hand in: Find the derivative of:
• HW: worksheet p.214 #7-26, 31, 32, 35
3.7 Chain Rule Practice Day 2Tues Nov 11
• Do Now
• Use the chain rule to find dy/dx
• 1)
•
• 2)
HW Review worksheet 214
• 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
• 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
Closure
• Journal Entry: If you had to describe how to use the chain rule to your friend, how would you?
• HW: Finish worksheet