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

2. Warm Ups: 1.Find the derivative of f(x) = (3x - 5) 2 using the power rule. f(x) = (3x - 5)(3x - 5) f(x) = 9x 2 - 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 - 5g’(x) = 3 h(x) = 3x - 5h’(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. Without the Chain Rule With the Chain Rule They are like onions, they have layers

4. 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 = u n then y’ = n·u n-1 · u’ OnionOutside Inside If y = (3x + 2) 5 then y’ = 5(3x + 2) 4 · 3 unun u5u5 nu n-1 u’u’ 5u4u4 3

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

7. Example 3: 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

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

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10. Example 5: Product Property Distribute x 2 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) Derivative Slope Equation of Tangent Line

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