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Basic Differentiation Rules and Rates of Change

Basic Differentiation Rules and Rates of Change . Section 2.2. The Constant Rule. Examples:. y = 5, find (dy/dx) f(x) = 13, find f’(x) y = (k π )/2 where k is an integer, find y’. The Power Rule. Examples:. f(x) = x 4 , find f’(x) g(x) = , find g’(x) , find.

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Basic Differentiation Rules and Rates of Change

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  1. Basic Differentiation Rules and Rates of Change Section 2.2

  2. The Constant Rule

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

  4. The Power Rule

  5. Examples: • f(x) = x4, find f’(x) • g(x) = , find g’(x) • , find

  6. The Constant Multiple Rule

  7. Examples: FunctionDerivative • y = 3/x • f(t) = (3t2)/7 • y = 5 • y = • y = (-5x)/3

  8. Example: Find the slope of the graph of f(x) = x7 when a. x = -2 b. x = 5

  9. Example: Find the equation of the tangent line to the graph of f(x) = x3, when x = -3

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

  11. Derivatives of Sine and Cosine • Examples: • y = 3 cos x • y = • y = 2x2 + cos x

  12. Extra Examples: Find each derivative:

  13. 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 = -16t2 + 100. Find the average velocity over each time interval: a. [1, 2] b. [1, 1.5] c. [1,1.1]

  14. 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) = -16t2+16t + 32 • When does the diver hit the water? • What is the diver’s velocity at impact?

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