1. Objectives: 1.Be able to find the derivative of functions by applying the Product Rule. Critical Vocabulary: Derivative, Tangent Daily Warm-Up: Find the derivative of the following functions 1. h(x) = (3x - 2x 2 )(5 + 4x) 2. h(x) = x 2 (3x - 2)(x 2 - 4x) h’(x) = -24x 2 + 4x + 15 h’(x) = 15x 4 - 56x 3 + 24x 2

2. I. The Product Rule The Product Rule is used when two or more functions are being multiplied together. h(x) = f(x) g(x) Look back at our warm-up problem: f(x) = (3x - 2x 2 )(5 + 4x) This could be done using the PRODUCT RULE!!!!! Example 1: Find the derivative: h(x) = (3x - 2x 2 )(5 + 4x) f(x) = 3x – 2x 2 f’(x) = 3 – 4x g(x) = 5 + 4x g’(x) = 4 h’(x) = (3x - 2x 2 )4 + (5 + 4x)(3 - 4x) h’(x) = 12x - 8x 2 + 15 - 8x - 16x 2 h’(x) = -24x 2 + 4x + 15

3. I. The Product Rule Example 2: Use the function to find the slope of the tangent line at the point (2, 3/2) f(x) = x -1 + 1f’(x) = -x -2 g(x) = x - 1g’(x) = 1 h’(x) = (x -1 + 1 )1 + (x - 1)(-x -2 ) h’(x) = x -1 + 1 - x -1 + x -2 f(x) = 5/4x - 1

4. I. The Product Rule Example 2: Use the function to find the slope of the tangent line at the point (2, 3/2) f(x) = 5/4x - 1

5. I. The Product Rule Example 3: Use the function to find the equation of the tangent line at the point (-1, 4): h(x) = 2x(x 2 + 3x) f(x) = 2xf’(x) = 2 g(x) = x 2 + 3x g’(x) = 2x + 3 h’(x) = (2x)(2x + 3) + (x 2 + 3x)(2) h’(x) = 4x 2 + 6x + 2x 2 + 6x h’(x) = 6x 2 + 12x h’(-1) = 6(-1) 2 + 12(-1) h’(-1) = 6 - 12 h’(-1) = -6 4 = -6(-1) + b 4 = 6 + b -2 = b f(x) = -6x - 2

6. I. The Product Rule Example 3: Use the function to find the equation of the tangent line at the point (-1, 4): h(x) = 2x(x 2 + 3x) h’(x) = 6x 2 + 12x h’(-1) = -6 f(x) = -6x - 2

7. I. The Product Rule The Product Rule can be extended to more than 2 functions. Example 4: Find the derivative: j(x) = x 2 (3x - 2)(x 2 - 4x ) f(x) = x 2 f’(x) = 2x g(x) = 3x - 2 g’(x) = 3 h’(x) = 2x(3x - 2)(x 2 - 4x) + x 2 (3)(x 2 - 4x) + x 2 (3x-2)(2x-4) h’(x) =6x 4 - 28x 3 + 16x 2 + 3x 4 - 12x 3 + 6x 4 – 16x 3 + 8x 2 h’(x) = 15x 4 - 56x 3 + 24x 2 h(x) = x 2 – 4xh’(x) = 2x - 4

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