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The Chain Rule

The Chain Rule. Section 3.5. The Chain Rule. According to Mrs. Armstrong … “Pull the chain and the light comes on!”. Introduction. Sludge Falls CO 2 is changing at rate of 0.02 ppm for each person Population growing at rate of 1000 people/yr

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The Chain Rule

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  1. The Chain Rule Section 3.5

  2. The Chain Rule • According to Mrs. Armstrong …“Pull the chain and the light comes on!”

  3. Introduction Sludge Falls • CO2 is changing at rate of 0.02 ppm for each person • Population growing atrate of 1000 people/yr • We seek rate of increasing pollution with respect to time (0.02 ppm/prsn)(1000 people/yr) = 20 ppm/yr

  4. Rate of Change of L with respect to t Rate of change of L with respect to P Rate of change of P with respect to t = A Composite Function The level of pollution L is a function of the population P, which is itself a function of time t. L = f(P(t)) Then L’ … is

  5. Population as a function of time Result in pollution as a function of time Pollution as a function of the population A Composite Function In Leibniz notation:

  6. The Chain Rule Given • y = f(u) is a differentiable function of u • u is also a differentiable function … of x • Then y = f(u(x)) • Then

  7. Example • Given:y = (6x3 – 4x + 7)3 • Then u(x) = 6x3 – 4x + 7and f(u) = u3 • Thusf’(x) = 3(6x3 – 4x + 7)2(18x2 – 4)

  8. Example • Given g(u) = u5 u(x) = 3x + 1 • Then g’(u) = ?? u’(x) = ?? • f(x) = (3x – 1)5f’(x) = ??

  9. Example • Find equation of tangent line to at (2,3)

  10. Try • Which is the u(x), the “inner” function? • Which is the f(u), the “outer” function? • What is u’(x), f’(u) ??

  11. Example • Try with multiple levels of nested functions

  12. Example • Try in combination with the product rule

  13. Assignment • Lesson 3.5 • Page 143 • Exercises 1 – 63 EOO (every other odd)

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