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Understanding the Chain Rule in Calculus: Mastering Composite Functions

This guide provides a comprehensive overview of the Chain Rule in calculus, showcasing its importance in finding derivatives of composite functions. Using clear examples and step-by-step explanations, we break down the concept of inner and outer functions. Key rules are highlighted, including how to handle specific derivatives involving absolute values. Whether you're struggling with algebra or looking to solidify your understanding of derivatives, this resource ensures you grasp the essentials of composite functions and the Chain Rule effectively.

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Understanding the Chain Rule in Calculus: Mastering Composite Functions

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  1. 2010 – Chain Rule AP CALCULUS

  2. If you can’t do Algebra . . . . I guarantee you someone will do Algebra on you!

  3. COMPOSITE FUNCTIONS Know: Need: Know: Need: REM: t(x)=1.0825(x) Order matters d(x)=.5(x) 10.83 10 Shirt for $20 10 20

  4. f(x) COMPOSITE FUNCTIONS f(g(x)) g(x) - what is done first. g(x) x g(x) Let and y = outside u = inside function function

  5. DECOMPOSE y = (outside) u = (inside). COMPOSITE FUNCTIONS

  6. Derivative of __________________________________________________ a composite function 2 functions 2 parts Texas two step In Words: __________________________________________________________ d(outside)leave the inside alone*d(inside)

  7. Chain Rule Leibniz Notation: In Words: __________________________________________________________ d(outside)leave the inside alone*d(inside)

  8. Example 1: Example: Let y = u = _______

  9. Example 2: Two steps:

  10. Example 3:

  11. Example 4: Example: Note:

  12. Example 6:

  13. Example 7: d(outside)*d(middle)*d(inside) Extended Chain: Ex: OR WORDS: Extended Chain:___________ Number of functions = number of parts

  14. Derivative of the Absolute Value Function REM: Do not simplify. Use the Chain Rule.

  15. General Rules: Working with number values Find the derivative. 1) f(x) + g(x) at x = 32) 2f(x) – 3g(x) at x =2 3) f(x)*g(x) at x = 24) f(x) / g(x) at x = 3 5) f(g(x)) at x = 26) (f(x))3 at x = 3 =

  16. Last Update • 10/13/07 • Assignment p. 153 # 13 – 31 odd, 56

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