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Objectives: 1. Be able to find the derivative of function by applying the Chain Rule

Chain Rule. Objectives: 1. Be able to find the derivative of function by applying the Chain Rule. Critical Vocabulary: Chain Rule. Warm Ups: Find the derivative of f(x) = (3x - 5) 2 using the power rule. Find the derivative of f(x) = (3x - 5) 2 using the product rule. Warm Ups:

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Objectives: 1. Be able to find the derivative of function by applying the Chain Rule

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  1. Chain Rule Objectives: 1. Be able to find the derivative of function by applying the Chain Rule Critical Vocabulary: Chain Rule • Warm Ups: • Find the derivative of f(x) = (3x - 5)2 using the power rule. • Find the derivative of f(x) = (3x - 5)2 using the product rule.

  2. Warm Ups: • Find the derivative of f(x) = (3x - 5)2 using the power rule. f(x) = (3x - 5)(3x - 5) f(x) = 9x2 - 30x + 25 f’(x) = 18x - 30 2. Find the derivative of f(x) = (3x - 5)2 using the product rule. f(x) = (3x - 5)(3x - 5) g(x) = 3x - 5 g’(x) = 3 h(x) = 3x - 5 h’(x) = 3 f’(x) = (3x - 5)(3) + (3x - 5)(3) f’(x) = 9x – 15 + 9x - 15 f’(x) = 18x – 30

  3. The Chain Rule: The chain rule deals with the composition of functions. With the Chain Rule Without the Chain Rule They are like onions, they have layers

  4. The Chain 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 Composition Function Idea: If y = un then y’ = n·un-1 · u’ If y = (3x + 2)5 then y’ = 5(3x + 2)4 · 3 Onion Outside un n un-1 u’ u5 5 u4 3 Inside

  5. The Chain 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 Example 1: f(x) = (3x - 5)2 h(u) = u2 h’(u) = 2u g(x) = 3x - 5 g(x) = 3 f’(x) = 2(3x - 5)• 3 f’(x) = 6(3x - 5) f’(x) = 18x - 30

  6. The Chain 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 Example 2: f(x) = (x2 + 1)3 h(u) = u3 h’(u) = 3u2 g(x) = x2 + 1 g(x) = 2x f’(x) = 3(x2 + 1)2 • 2x f’(x) = 6x(x2 + 1)2

  7. The Chain 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 Example 3:

  8. Example 4: Find the equation of the tangent line when x = 2 First lets find the y-value f(x) = mx + b

  9. Page 290 #7-23 odds

  10. Example 5: Product Property Distribute x2 Factor GCF Distribute 2 Combine Like Terms No Negative Exponents

  11. Example 6:

  12. Example 7: Find the equation of the tangent line at (2, 2) Slope Equation of Tangent Line Derivative

  13. Page 290 - 291 #25 - 35 odds

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