1. Objectives: 1.Be able to find the derivative using the Constant Rule. 2.Be able to find the derivative using the Power Rule. 3.Be able to find the derivative using the Constant Multiple Rule. 4.Be able to find the derivative using the Sum and Difference Rules. Critical Vocabulary: Constant Rule, Power Rule, Constant Multiple Rule, Sum and Difference Rules.

2. I. The Constant Rule Example: Find the derivative of f(x) = 3 using the definition f(x) = 3 f(x + Δx) = 3 The derivative of a constant function is zero. if c is a real number Example 1: Find the derivative of f(x) = 6 Example 2: Find the derivative of f(x) = -8

3. II. The Power Rule Example: Find the derivative of each function using the definition 1. f(x) = 3x 2 2. f(x) = 4x 2 3. f(x) = 5x 2 1. f’(x) = 6x 2. f’(x) = 8x 3. f’(x) = 10x 1. f(x) = 3x 3 2. f(x) = 4x 3 3. f(x) = 5x 3 1. f’(x) = 9x 2 2. f’(x) = 12x 2 3. f’(x) = 15x 2 What kind of patterns do you observe?

4. II. The Power Rule If n is a rational number, then the function f(x) = x n is differentiable and For f to be differentiable at x = 0, n must be a number such that x n-1 is defined on an interval containing zero Example 1: Find the derivative of f(x) = x Example 2: Find the derivative of f(x) = x 6

5. II. The Power Rule If n is a rational number, then the function f(x) = x n is differentiable and For f to be differentiable at x = 0, n must be a number such that x n-1 is defined on an interval containing zero Sometimes, you need to rewrite an expression if it is not in the form x n (Your final answer may not contain negative exponents) Example 3: Find the derivative of

6. II. The Power Rule If n is a rational number, then the function f(x) = x n is differentiable and For f to be differentiable at x = 0, n must be a number such that x n-1 is defined on an interval containing zero Sometimes, you need to rewrite an expression if it is not in the form x n (Your final answer may not contain negative exponents) Example 4: Find the derivative of

7. II. The Power Rule If n is a rational number, then the function f(x) = x n is differentiable and For f to be differentiable at x = 0, n must be a number such that x n-1 is defined on an interval containing zero Sometimes, you need to rewrite an expression if it is not in the form x n (Your final answer may not contain negative exponents) Example 5: Use the function f(x) = x 2 to find the slope of the tangent line at the point (2, 4). General Rule Slope: 4

8. III. The Constant Multiple Rule If f is a differentiable function and c is a real number, then cf is differentiable and Example 1: Find the derivative of Example 2: Find the derivative of

9. III. The Constant Multiple Rule If f is a differentiable function and c is a real number, then cf is differentiable and Example 3: Find the derivative of

10. IV. The Sum and Difference Rules Example 1: Find the slope of the tangent line at (1, -1) of f(x) = x 3 - 4x + 2 The sum (or difference) of two differentiable functions is differentiable and is the sum (or difference) of their derivatives. f’(x) = 3x 2 - 4 m = -1

11. IV. The Sum and Difference Rules Example 2: Find the equation of the tangent line to: f(x) = -½x 4 + 3x 3 – 2x at (-1, -3/2) f’(x) = -2x 3 + 9x 2 – 2 m = -2(-1) 3 + 9(-1) 2 – 2 m = 2 + 9 – 2 m = 9 f(x) = mx + b -3/2 = 9(-1) + b -3/2 = -9 + b 15/2 = b f(x) = 9x + 15/2

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13. V. Additional Examples Example 1: Find all the points at which the graph of f(x) = x 3 – 3x has horizontal tangent lines. f’(x) = 3x 2 - 3 3x 2 - 3 = 0 3x 2 = 3 x 2 = 1 x = 1 and x = -1 (1,-2) and (-1, 2)

14. V. Additional Examples Example 2: Find all the points at which the graph of f(x) = x 4 – 4x + 5 has horizontal tangent lines. f’(x) = 4x 3 - 4 4x 3 - 4 = 0 4x 3 = 4 x 3 = 1 x = 1 (1, 2)

15. V. Additional Examples Example 3: Find k such that the line is tangent to the graph of the function. Function: f(x) = k – x 2 Tangent: f(x) = -4x + 7 Equate Functions: k – x 2 = -4x + 7 Equate Derivatives: -2x = -4 x = 2 k – (2) 2 = -4(2) + 7 k – 4 = -1 k = 3

16. Part 1: Page 272-273 #3-47 odds Part 2: Page 272-273 #49 – 56 all Worksheet 4.2A