52 Find a formula for d dx f 3 x 2 assuming f is differentiable 53 What is the

52 find a formula for d dx f 3 x 2 assuming f is

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52. Find a formula for d dx f (3 x 2 ) assuming f is differentiable. 53. What is the 45th derivative of y = sin( x )? 54. Find a formula for f ( n ) ( x ) if f ( x ) = xe x . Recall that f ( n ) denotes the n th derivative of f . 55. Find a formula for g ( n ) ( x ) if g ( x ) = 1 x . Recall that g ( n ) denotes the n th derivative of g . 56. Find an equation of the tangent to the graph of y = f 0 ( x ) at x = 1 where f ( x ) = x 5 - 3 x 2 + x . 57. For each of the graphs below, provide a rough sketch of the derivative of the corresponding function. Related Rates 58. Suppose x and y are differentiable functions of t and are related by y = x 2 - 1. Find dy/dt when x = 2 given that dx/dt = 3. 59. A rock is dropped into a calm pond, causing ripples in the shape of concentric circles. The radius of the outer ripple is increasing at a rate of 12 cm/s. When the radius is 30 cm, at what rate is the total area of the outer ripple changing? 60. A spherical balloon is deflated so that its volume decreases at a constant rate of 3 in 3 / s. How fast is the balloon’s diameter decreasing when the radius is 2 inches? 61. A 10-foot ladder is leaning against a building. If the top of the ladder slides down the wall at a constant rate of 2 feet per second, how fast is the acute angle that the ladder makes with the ground decreasing when the top of the ladder is 5 feet from the ground? (Give your answer in radians per second.)
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MAT 136: Calculus I Exam 2 Supplemental Problems 62. Two cars start moving from the same point. One travels south at 60 miles per hour and the other travels west at 25 miles per hour. At what rate is the distance between the cars increasing two hours later? Additional Derivative Practice If you feel that you need additional practice with derivative rules, find the first derivative of each of the following functions. Power Rule 63. f ( x ) = x - x 3 64. y = 3 x 2 - x + 4 x + π 2 65. f ( x ) = 4 x 2 - x 2 4 66. h ( x ) = 3 x 67. f ( x ) = x 2 - e 2 68. g ( x ) = p x Chain Rule 69. f ( x ) = ( x 2 - 1) 10 70. f ( x ) = p 1 + 1 + 2 x 71. g ( x ) = (3 x 2 + 3 x - 6) - 8 72. f ( x ) = 4 9 - x Power, Product, Quotient, Chain Rules 73. h ( x ) = ( x - 4) 3 ( x + 4) 5 74. f ( x ) = x 3 x 2 - x 75. f ( x ) = (5 x 2 - 3)( x 2 - 2) x 2 + 2 76. g ( x ) = x x + 17 x 77. f ( t ) = 3 t 2 + 2 t 78. g ( w ) = w 3 ( w + 3) 5 79. h ( s ) = ( s - 2 ) 3 80. f ( x ) = 5 x 81. g ( x ) = 3 q 5 p x 82. m ( t ) = t 2 - 5 t 83. g ( y ) = q 1 + p 1 + y 84. h ( s ) = ( s + 1) 5 s - 1 85. f ( x ) = 2 x - 1 x + 1 86. f ( x ) = ( x + 2) 2 (3 x - 4 x 5 ) 100 (8 - x ) 7 Exponential Functions 87. f ( t ) = e 3 t 88. y = t 2 e t 3 89. g ( z ) = 2 3 3 z - z 2 90. h ( k ) = 7 e - 5 - 7 e - 5 k + k 2 ln( e 4 ) 91. i ( r ) = 2 4 r 92. A ( t ) = Pe rt ( P, r
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