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The Chain Rule

The Chain Rule. Wednesday, Feb. Velocity & Acceleration. Velocity = slope of a displacement-time graph Acceleration = slope of a velocity-time graph. Velocity & Acceleration. If the velocity is positive, The object is moving forward

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The Chain Rule

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  1. The Chain Rule Wednesday, Feb

  2. Velocity& Acceleration Velocity = slope of a displacement-time graph Acceleration = slope of a velocity-time graph

  3. Velocity& Acceleration If the velocity is positive, • The object is moving forward • The object might be moving forward depending on the sign of the acceleration • The object is moving backward • The object might be moving backward depending on the sign of the acceleration

  4. Velocity& Acceleration If the velocity is positive, • The object is moving forward • The object might be moving forward depending on the sign of the acceleration • The object is moving backward • The object might be moving backward depending on the sign of the acceleration

  5. Velocity& Acceleration If the velocity is negative, • The object is moving forward • The object might be moving forward depending on the sign of the acceleration • The object is moving backward • The object might be moving backward depending on the sign of the acceleration

  6. Velocity& Acceleration If the velocity is negative, • The object is moving forward • The object might be moving forward depending on the sign of the acceleration • The object is moving backward • The object might be moving backward depending on the sign of the acceleration

  7. Velocity& Acceleration If the acceleration is positive and the velocity is negative • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  8. Velocity& Acceleration If the acceleration is positive and the velocity is negative • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  9. Visualizing Positive Acceleration + velocity + acceleration − velocity + acceleration

  10. Visualizing Positive Acceleration + velocity − acceleration − velocity − acceleration

  11. Velocity& Acceleration If the acceleration is positive and the velocity is positive • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  12. Velocity& Acceleration If the acceleration is positive and the velocity is positive • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  13. Velocity& Acceleration If the acceleration is negative and the velocity is positive • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  14. Velocity& Acceleration If the acceleration is negative and the velocity is positive • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  15. Velocity& Acceleration If the acceleration is negative and the velocity is negative • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  16. Velocity& Acceleration If the acceleration is negative and the velocity is negative • The object is speeding up in the forwards direction • The object is speeding up in the backwards direction • The object is slowing down while moving forwards • The object is slowing down while moving backwards

  17. The Chain Rule Goal of today’s class: Learn how to differentiate a function inside a function.

  18. The Chain Rule For example: what is the tangent slope of: y = (4 – x2)¾ What is dy/dx? We can simplify this by making a substitution: Let R = (4 – x2) Then y = R¾ dydydR dx dR dx =

  19. The Chain Rule We can write the chain rule in leibniz notation: dydydR dx dR dx = Or lagrange notation: If f(x) = g(h(x)) Then f’(x) = g’(h(x)) h’(x)

  20. The Chain Rule: Example What is the derivative of 3(x3 – 5x)4?

  21. The Chain Rule & Product Rule: Example What is the derivative of 3x(x3 – 5x)4?

  22. Practice in your teams Take the derivative of the following functions: f(x) = √(x6 + ½x3) f(x) = 4x(5x2 – x)6 f(x) = 5(x - √x)3 + 6x2 + 9

  23. Practice Individually Page 117 #1, 4, 7, 8, 14, 17

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