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The Chain Rule

The Chain Rule. Objective. To use the chain rule for differentiation. The Chain Rule. The Chain Rule is a technique for differentiating composite functions . Composite functions are made up of layers of functions inside of functions. The Chain Rule. The outside. The Chain Rule.

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The Chain Rule

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  1. The Chain Rule

  2. Objective • To use the chain rule for differentiation.

  3. The Chain Rule • The Chain Rule is a technique for differentiating composite functions. • Composite functions are made up of layers of functions inside of functions.

  4. The Chain Rule The outside

  5. The Chain Rule Peel it away

  6. The Chain Rule There’s still more to peel.

  7. The Chain Rule Some functions have many layers that must be peeled away in order to find their derivatives.

  8. The Chain Rule

  9. The Chain Rule Inside function Outside function

  10. The Chain Rule

  11. The Chain Rule 1. Identify inner and outer functions. 2. Derive outer function, leaving the inner function alone. 3. Derive the inner function.

  12. The Chain Rule (Example A)

  13. The Chain Rule • Key Point: If the inside function contains something other than plain old “x,” you must use the Chain Rule to find the derivative. Inside function

  14. The Chain Rule Outside function

  15. The Chain Rule Inside function

  16. The Chain Rule Outside function

  17. The Chain Rule

  18. The Chain Rule The derivative of the outside The derivative of the inside (blop)

  19. The Chain Rule

  20. The Chain Rule CAUTION: Possible mistakes ahead! What mistake did I make? I changed the inside function and did not multiple by the derivative of the inside function.

  21. The Chain Rule (Example B)

  22. The Chain Rule (Example C)

  23. The Chain Rule (Example D)

  24. The Chain Rule (Example E)

  25. The Chain Rule (Example F)

  26. The Chain Rule (Example G)

  27. The Chain Rule (Example H)

  28. The Chain Rule (Example I)

  29. Conclusion • Remember: When a function is inside another function, use the Chain Rule to find the derivative. • First, differentiate the outside function, leaving the inside function alone. • Last, differentiate the inside function.

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