Calculus/Differentiation/Basics of Differentiation/Solutions

Find the Derivative by Definition 1. 2. 3. 4. 5. 6. 7. 8. 9. Prove the Constant Rule 10. Use the definition of the derivative to prove that for any fixed real number , Find the Derivative by Rules Power Rule 11. 12. 13. 14. 15. 16. 17. 18. 19. Product Rule 20. 21. 22. 23. Quotient Rule 29. 30. 31. 32. 33. 34. 35. Chain Rule 43. Let . Then 44. Let . Then 45. Let . Then 46. Let . Then 47. Let . Then 48. Let . Then 49. Let . Then 50. Let . Then 51. Let . Then 52. Let . Then 53. Let . Then Exponentials 54. 55. Let . Then 56. Let Then Using the chain rule, we have The individual factor are So 57. Logarithms 58. 47. 59. Let . Then 60. 61. Trigonometric functions 62. 64. More Differentiation 65. 66. 67. Let . Then 68. 69. 70. 71. 72. 73. Implicit Differentiation Use implicit differentiation to find y' 74. 75. Logarithmic Differentiation Use logarithmic differentiation to find : 76. 77. 78. 79. 80. Equation of Tangent Line For each function, , (a) determine for what values of the tangent line to is horizontal and (b) find an equation of the tangent line to at the given point. 81. : a) b) 82. : a) b) 83. : a) b) 84. : a) b) 85. : a) b) 86. : a) / b) 87. Find an equation of the tangent line to the graph defined by at the point (1,-1). 88. Find an equation of the tangent line to the graph defined by at the point (1,0). Higher Order Derivatives 89. What is the second derivative of ? 90. Use induction to prove that the (n+1)th derivative of a n-th order polynomial is 0. base case: Consider the zeroth-order polynomial, . induction step: Suppose that the n-th derivative of a (n-1)th order polynomial is 0. Consider the n-th order polynomial, . We can write where is a (n-1)th polynomial. Advanced Understanding of Derivatives 91. Let be the derivative of . Prove the derivative of is . Suppose . Let . Therefore, if is the derivative of , then is the derivative of . 92. Suppose a continuous function has three roots on the interval of . If , then what is ONE true guarantee of using (a) the Intermediate Value Theorem; (b) Rolle's Theorem; (c) the Extreme Value Theorem. These are examples only. More valid solutions may exist. (a) is continuous. Ergo, the intermediate value theorem applies. There exists some such that , where . (b) Rolle's Theorem does not apply for a non-differentiable function. (c) is continuous. Ergo, the extreme value theorem applies. There exists a so that for all . 93. Let , where is the inverse of . Let be differentiable. What is ? Else, why can not be determined? If , then . We can use implicit differentiation. 94. Let where is a constant. Find a value, if possible, for that allows each of the following to be true. If not possible, prove that it cannot be done. (a) The function is continuous but non-differentiable. (b) The function is both continuous and differentiable. (a) . . However, for , we find that , so makes the function continuous but non-differentiable. (b) There is no that allows the function to be differentiable and continuous. A proof of this is simple. However, To allow the best possible chance, we will let : For any other , one will have an infinity on the left-hand sided limit. Therefore, there is no possible that allows the function to be differentiable and continuous.