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Basic Differentiation Rules and Rates of Change Section 2.2

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The Constant Rule

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Examples: 1. y = 5, find (dy/dx) 2. f(x) = 13, find f’(x) 3. y = (kπ)/2 where k is an integer, find y’

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The Power Rule

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Examples: 1. f(x) = x 4, find f’(x) 2. g(x) =, find g’(x) 3., find

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The Constant Multiple Rule

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Examples: FunctionDerivative y = 3/x f(t) = (3t 2 )/7 y = 5 y = y = (-5x)/3

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Example: Find the slope of the graph of f(x) = x 7 when a. x = -2b. x = 5

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Example: Find the equation of the tangent line to the graph of f(x) = x 3, when x = -3

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The Sum and Difference Rules Sum Rule Difference Rule Example: f(x) = Find f’(x)

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Derivatives of Sine and Cosine Examples: 1. y = 3 cos x 2. y = 3. y = 2x 2 + cos x

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Extra Examples: Find each derivative:

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Rates of Change: Average Velocity Velocity (Rate) = If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function s = -16t 2 + 100. Find the average velocity over each time interval: a. [1, 2] b. [1, 1.5] c. [1,1.1]

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Instantaneous Velocity: If s(t) is the position function, the velocity of an object is given by. Velocity can be. Speed is the of the velocity. Example: At time t = 0, a diver jumps from a board that is 32 feet above the water. The position of the diver is given by s(t) = -16t 2 +16t + 32 a. When does the diver hit the water? b. What is the diver’s velocity at impact?

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