unj fmipa-fisika

1. Kalkulus II Teguh Budi P, M.Si Sesion#25-26 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam

2. Partial Derivatives The Chainrule 2 Outline 1/9/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

3. Multivariable Functions and Their Derivatives(part 3) 3 1/9/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

4. z 100 An Introduction to Partial Derivatives 10 y 10 x 1/9/2011 4 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. When we have functions with more than one variable, we can find partial derivatives by holding all the variables but one constant. z 100 10 y 10 x (eff sub ecks) Note: is also written as 1/9/2011 5 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. z would give you the slope of the tangent in the plane y=0 or in any plane with constant y. 100 10 y 10 x In other words, how is changing one variable going to change the value of the function? 1/9/2011 6 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

7. Mixed variables are also possible: Both answers are the same! 1/9/2011 7 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. The Chain Rule Consider a simple composite function: 1/9/2011 8 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. and another: 1/9/2011 9 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

10. This pattern is called the chain rule. and one more: 1/9/2011 10 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. If is the composite of and , then: Find: example: Chain Rule: 1/9/2011 11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

12. We could also do it this way: 1/9/2011 12 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

13. Here is a faster way to find the derivative: Differentiate the outside function... …then the inside function 1/9/2011 13 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

14. Another example: It looks like we need to use the chain rule again! derivative of the outside function derivative of the inside function 1/9/2011 14 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

15. The chain rule can be used more than once. Another example: (That’s what makes the “chain” in the “chain rule”!) 1/9/2011 15 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

16. The formulas on the memorization sheet are written with instead of . Don’t forget to include the term! Derivative formulas include the chain rule! etcetera… 1/9/2011 16 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

17. The most common mistake on the chapter 3 test is to forget to use the chain rule. Every derivative problem could be thought of as a chain-rule problem: The derivative of x is one. derivative of outside function derivative of inside function 1/9/2011 17 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

18. Divide both sides by The slope of a parametrized curve is given by: The chain rule enables us to find the slope of parametrically defined curves: 1/9/2011 18 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

19. Example: These are the equations for an ellipse. 1/9/2011 19 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

20. Thank You 1/9/2011 20 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |