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**Unformatted text preview: **Math 1007, Fall 2006 (These slides replace neither the text book nor the lectures.) Part III: Differentiation 16. Differentiation: Four Basic Rules (33-37) 17. Product Rule and Quotient Rule (38-40) 18. The Derivative (41-44) 19. The Chain Rule (45-50) 20. Derivatives of Trigonometric Functions (51-54) 21. Derivatives of Exponential Functions (55-56) 22. Derivatives of Logarithmic Functions (57) 23. Implicit Differentiation (58-63) 24. Derivatives of Inverse Trig Functions (64-65) 25. Higher Order Derivatives (66-70) 26. Logarithmic Differentiation (71-74) 32 16. Differentiation: Four Basic Rules Rule 1: Derivative of a Constant d dx ( c ) = 0 ( c a constant) Rule 2: The Power Rule If n is any real number, then d dx ( x n ) = nx n- 1 Example 1 f ( x ) = x f ( x ) = d dx ( x ) = 1 x 1- 1 = x = 1 Example 2 f ( x ) = x 3 / 2 f ( x ) = d dx ( x 3 / 2 ) = 3 2 x 1 / 2 33 Rule 3: Derivative of a Constant Multiple of a Function d dx [ cf ( x )] = c d dx f ( x ) ( c a constant) Example 3 f ( x ) = 2 x 3 f ( x ) = d dx (2 x 3 ) = 2 d dx x 3 = 2 · (3 x 2 ) = 6 x 2 Rule 4: The Sum Rule d dx [ f ( x ) ± g ( x )] = d dx [ f ( x )] ± d dx [ g ( x )] Example 4 f ( x ) = 4 x 5 + 10 x 2 f ( x ) = d dx (4 x 5 ) + d dx 10 x 2 = 20 x 4 + 20 x 34 Problem: Find the slope and an equation of the tangent line to the graph of f ( x ) = 2 x + 1 √ x at the point (1 , 3). Solution f ( x ) = 2 x + 1 √ x = 2 x + x- 1 / 2 f ( x ) = 2- 1 2 x- 3 / 2 = 2- 1 2 x 3 / 2 f (1) = 2- 1 2(1) 3 / 2 = 2- 1 2 = 3 2 Using slope m = 3 2 and point ( x, y ) = (1 , 3) y = mx + b 3 = 3 2 (1) + b b = 3- 3 2 = 3 2 = ⇒ y = 3 2 x + 3 2 Hence the slope of the tangent is 3 / 2 and the equation of the tangent line is y = 3 2 x + 3 2 . 35 Application: Altitude of a Rocket The altitude of a rocket (in feet) t seconds into flight is s = f ( t ) =- t 3 + 96 t 2 + 195 t + 5 ( t ≥ 0) . Find the initial velocity of the rocket. Solution: Velocity is the derivative of Altitude The rocket’s velocity at any time t is v = f ( t ) =- 3 t 2 + 192 t + 195 Since f (0) = 195, the rocket’s initial velocity is 195 ft/second. 36 Application: Population Growth A study of Sunbelt projects that the town’s population in the next 3 years will grow according to P ( t ) = 50 , 000 + 30 t 3 / 2 + 20 t, where P ( t ) denotes the population t months from now. (a) How fast will the town’s population be increasing in 9 months? (b) in 16 months? Solution: The change in population is the derivative of population P ( t ) = 3 2 30 t 1 / 2 + 20 = 45 t 1 / 2 + 20 (a) In 9 months, the population change is P (9) = 45(9) 1 / 2 = 45 · 3 + 20 = 155 As this value is positive , the population is increasing by 155 people/month. (b) In 16 months, the population change is P (9) = 45(16) 1 / 2 = 45 · 4 + 20 = 200 , so the population is increasing by 200 people/month....

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