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Section 2.4 – The Chain Rule

Section 2.4 – The Chain Rule. Warm-Up. Wrong answer: The exponent of 30. Although the exponent makes the derivative difficult, we could use the product rule to find the derivative. Right answer: The function inside of the secant function.

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Section 2.4 – The Chain Rule

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  1. Section 2.4 – The Chain Rule

  2. Warm-Up Wrong answer: The exponent of 30. Although the exponent makes the derivative difficult, we could use the product rule to find the derivative. Right answer: The function inside of the secant function. We have not taken the derivative of a composition of functions. Explain why we can not differentiate the function below:

  3. Composition of Functions COMPOSITION OF FUNCTIONS If and , find .

  4. Decomposition of Functions If each function below represents , define and . DECOMPOSITION OF FUNCTIONS

  5. The Chain Rule Other ways to write the Rule: If y = f(u)is a differentiable function of u and u = g(x)is a differentiable function of x, then y = f(g(x)) is a differentiable function of x, and

  6. Instructions for The Chain Rule Make sure each function can be differentiated. For , to find : • Decompose the function • Differentiate the MOTHER FUNCTION • Differentiate the COMPOSED FUNCTION • Multiply the resultant derivatives • Substitute for u and Simplify

  7. Example 1 Define f and u: Find the derivative of f and u: Use the Chain Rule: Substitute: Substitute for u: Simplify: Find if and .

  8. Example 2 Define f and u: Find the derivative of f and u: Use the Chain Rule: Substitute: Substitute for u: Simplify: Differentiate .

  9. Example 3 Define h and u: Find the derivative of h and u: If f and g are differentiable, , , and ; find .

  10. Example 4 Define f and u: Find the derivative of f and u: Find if .

  11. White Board Challege Find f '(-2) if:

  12. Example 5 Define f and u: Find the derivative of f and u: OR Differentiate .

  13. Example 6 Use the old derivative rules Chain Rule Twice Differentiate .

  14. Example 7 Quotient Rule Chain Rule Find the derivative of the function .

  15. Example 8 Product Rule Chain Rule Twice Differentiate .

  16. White Board Challege Find the equation of the tangent to the curve y=3sin(2x) at the point:

  17. Example 9 Chain Rule Chain Rule Again Differentiate .

  18. Example 10 Find the Derivative Evaluate the Derivative at x = π Find the equation of the line Find an equation of the tangent line to at .

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