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Finding the Derivative by the Limit
Process In Exercises 15–28, find the derivative
of the function by the limit process
f (x) = −5x
Step by step
Solved in 2 steps
- In Exercises 75–78, sketch the graph of a function y = f(x) that satis- fies the given conditions. No formulas are required-just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) 75. f(0) = 0, f(1) = 2, f(-1) = -2, lim f(x) = -1, and x--00 lim f(x) = 1 76. f(0) = 0, lim f(x) = 0, lim f(x) = 2, and lim f(x) = -2 x→0* %3D 77. f(0) = 0, lim f(x) = 0, lim f(x) = lim f(x) = ∞, x-too x→1- x--1+ = -0, and lim f(x) = -∞ lim f(x) x→1* 78. f(2) = 1, f(-1) = 0, lim f(x) = 0, lim f(x) = ∞, x→0* lim f(x) = -00, and lim f(x) = 1 X -00Mixed Partial DerivativesIn Exercises 55–60, verify thatPiecewise-Defined FunctionsGraph the functions in Exercises 25–28.
- Decomposition of a Composite FunctionIn Exercises 5–12, complete the table. y=(6 x-5)^{4}Precise Definition of Limit In Exercises 7–10, use the formal definition of limit to prove that the function is continuous at c.Finding a Derivative In Exercises 13–32, findthe derivative of the function. y = 5(2 − x3)4
- Finding a Derivative In Exercises 7–26, usethe rules of differentiation to find the derivative ofthe function. \text { 16. } g(x)=6 x+3In Exercises 11–18, graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. 11. f(x) = 4" 13. g(x) = ()* 15. h(x) = (})* 17. f(x) = (0.6) 12. f(x) = 5" 14. g(x) = () 16. h(x) = (})* 18. f(x) = (0.8)* %3!Exercises 63–86: Use transformations to sketch a graph of f. 63. f(x) = x² – 3 64. f(x) = -x² 65. f(x) = (x = 5)² + 3 66. f(x) = (x + 4)° 67. flx) = -Vx 68. f(x) = 2(x = 1F + 1 69. f(x) = -x² + 4 70. f(x) = V=x 71. f(x) = |x| – 4 73. f(x) = Vx – 3 + 2 74. f(x) = |x + 2| – 3 72. flx) = Vx + 1 76. flx) = |x| 78. f(x) = 2Vx – 2 - 1 75. f(x) = |2x| 77. f(x) = 1 – Vx 79. f(x) = -Vī - x 81. f(x) = V-(x + 1) 80. f(x) = V-x – 1 82. f(x) = 2 + V-(x – 3) 83. f(x) = (x = 1) 84. f(x) = (x + 2) 85. f(x) = -x' 86. f(x) = (-x)' + 1
- The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36.Finding a Derivative In Exercises 13–32, findthe derivative of the function. y=\frac{1}{x-2}In Exercises 73–78, the graph of f is shownin the figure. Sketch a graph of the derivative of f. To print anenlarged copy of the graph, go to MathGraphs.com.image5