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Mastering the Chain Rule in Calculus: Composite Functions Explained

Explore the intricacies of the Chain Rule in calculus, particularly focusing on composite functions. This guide covers essential concepts including how to decompose functions, apply Leibniz notation, and use the Chain Rule effectively for differentiation. With various examples, including the derivative of absolute value functions, teachers and students alike will gain clarity on handling composite functions. Plus, follow general rules for finding derivatives with number values. Perfect for anyone struggling with algebra and advanced calculus concepts!

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Mastering the Chain Rule in Calculus: Composite Functions Explained

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  1. 2011 – 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: f(x)=1.0825(x) g(x)=.5(x) 10 10 20

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

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

  6. Derivative of __________________________________________________ In Words: __________________________________________________________

  7. Chain Rule Leibniz Notation: In Words: __________________________________________________________

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

  9. Example 2:

  10. Example 3:

  11. Example 4: Example: Note:

  12. Example 6:

  13. Example 7: Extended Chain: Ex: OR WORDS: Extended Chain:___________

  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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