CONTINUITY AND DIFFERENTIABILITY
The derivative of logx. w.r.t. x is. 1 x. ; i.e.. 1. (log ) d x dx x. = . 5.1.12 Logarithmic differentiation is a powerful technique to differentiate
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6.2 Properties of Logarithms
(Inverse Properties of Exponential and Log Functions) Let b > 0 b = 1. • ba = c if and only if logb(c) = a. • logb (bx) = x for all x and blogb(x) = x for
S&Z . & .
Appendix: algebra and calculus basics
Sep 28 2005 6. The derivative of the logarithm
algnotes
New sharp bounds for the logarithmic function
Mar 5 2019 In this paper
DIFFERENTIAL EQUATIONS
An equation involving derivative (derivatives) of the dependent variable with Now substituting x = 1 in the above
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Class 6 Notes
Sep 24 2018 log x. A more refined answer: it looks like a certain integral
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Answers to Exercises
Product 10 - 15 Hence to find the n-th derivative we just divide n by 4
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Dimensions of Logarithmic Quantities
to imply that log (x) is itself dimensionless whatever the Thus d log (x) is always dimensionless
Appendix Algebra and Calculus Basics
The derivative of the logarithm d( log x)/dx
Chapter 8 Logarithms and Exponentials: logx and e
x. = d dx log x. Then log xy and log x have the same derivative from which it follows by the Corollary to the Mean Value Theorem that these two functions
chapter
Answers to Exercises
I am much indebted to Anthony Petrello for some of the answers to the exer cises.I, §2, p. 13
1. -3 < x < 3 2. - 1 x
0 3. -j3 x -1 or 1 x j3
4. x < 3 or x > 7 5. - 1 < x < 2 6. x < -lor
x > 1 7. - 5 < x < 58. - 1 x 0 9. x 1 or x = 0 10. x -10 or x = 5
11. x -10 or x = 5 12. x 1 or x = -! 13. x < -414. -5 17. -2 < x < 8 18. 2 < x < 4 19. -4 < x < 10 20. x < -4 and x> 10
21. x < -10 and x > 4
I, §3, p. 17
1. 2. (2x 1) 3. 0, 2, 108 4. 2z -Z2, 2w - w
2 5. x # j2 or -j2. f(5) = b 6. All x. f(27) = 3
7. (a)
1 (b) 1 (c) -1 (d) -1 8. (a) 1 (b) 4 (c) 0 (d) 0
9. (a) -2 (b) -6 (c) x2+4x-2 10. 2
II. (a) odd (b) even (c) odd (d) odd
I, §4, p. 20
1. 8 and 9 2.! and -1 3. /6 and 2 4. and 2
1/3 5. -h and! 6. 9 and 8
7. -! and -1 8.! and! 9. 1 and -! 10. -5:2 and!
A2 ANSWERS TO EXERCISES
II. Yes. Suppose a is negative, so write a = -b where b is positive. Let c be a positive number such that c" = b. Then (-c)" = a because (-1)" = -1 since n is odd. II, §1, p. 24
3. x negative, y positive 4. x negative, y negative
II, §3, p. 33
5. y = + i 6. y = -ix + S 7. x = J2
9 9j3 8. y = --;:;-x + 4 ---;:;-9. y = 4x -3 10. y = -2x + 2
",3+3 ",3+3 II. y= -!x+3+ J'!-12. y=j3x+S+j3 19. -! 20.-8
21. 2 + j2 22.!<3 + j3) 23. y = (x -n>(J22_ n) + 1
24. 25. y= -(x+ l)(j23+ 1)+2
26. y = (x + 1)(3 + j2) + j2 29. (a) x = -4, y = -7 (b) x = y = i
(c) x = -!, y = 1 (d) x = -6, Y = -5 II, §4, p. 35
1.J97 2.j2 3.Js2 4.Ji3 5.!j5 6.(4,-3) 7. SandS 8.(-2,5)
9. Sand 7
II, §8, p. 51
5. (x -2)2 + (y + 1)2 = 25 6. x
2 + (y -
1)2 = 9 7. (x + \)2 + y2 = 3
8. y + ¥ = 2(x + W 9. y - 1 = (x + 2)2 10. y + 4 = (x -\)2
II. (x + 1)2 + (y -2)2 = 2 12. (x -2)2 + (y -\)2 = 2 13. x + ¥-= 2(y + !)2 14. x-I = (y + 2)2
III, §1, p. 61
1. 4 2. -2 3. 2 4. i S. -! 6. 0 7. 4 8. 6 9. 3 10. 12 II. 2
12. 3 13. a
ANSWERS TO EXERCISES
III, §2, p. 70
Tangent line at x = 2 Slope at x = 2
1. 2x y = 4x - 3 4
2. 3x2 y = 12x -16 12
3. 6x2 y = 24x -32 24
4.6x y = 12x -12 12
5.2x y = 4x - 9 4
6. 4x + I Y = 9x - 8 9
7. 4x -3 Y = 5x - 8 5
3x2 8. 2+2 y = 8x - 8 8
9. Y = + 1 -(x + 1)2 -9 2 Y = +.!j
2 10. - (x + 1)2 -9 III, §3, p. 75
2 I 1 I. 4x + 3 2. -2 3. 2 4. 2x + 1 5. -2 6. 9x
2 (2x + 1) (x + 1) (2x -1) 3x 2 10. 2 + I 11. _2/X2 12. -3/x
2 13. -2/(2x -W 14. -3/(3x + 1)2 15. -1/(x + 5)2 16. -I/(x _ 2)2
17. -2x-3 18. -2(x + 1)-3
III, §4, p. 78
1. X4 + 4x
3h + 6x 2 h 2 + 4xh3 + h4 2. 4x3 3. (a) jx-
I/3 (b) (c) 4. Y = 9x -8 5. Y = tx + t slope t -3 7 -3 1 -fi 1 6. Y = Y x + 32' slope y 7. Y = 2-fi x + 2' slope 2-fi
8. (a) !5-
3/4 (b) _!r
Answers to Exercises
I am much indebted to Anthony Petrello for some of the answers to the exer cises. I, §2, p. 13
1. -3 < x < 3 2. - 1 x
0 3. -j3 x -1 or 1 x j3
4. x < 3 or x > 7 5. - 1 < x < 2 6. x < -lor
x > 1 7. - 5 < x < 5 8. - 1 x 0 9. x 1 or x = 0 10. x -10 or x = 5
11. x -10 or x = 5 12. x 1 or x = -! 13. x < -4 14. -5 17. -2 < x < 8 18. 2 < x < 4 19. -4 < x < 10 20. x < -4 and x> 10
21. x < -10 and x > 4
I, §3, p. 17
1. 2. (2x 1) 3. 0, 2, 108 4. 2z -Z2, 2w - w
2 5. x # j2 or -j2. f(5) = b 6. All x. f(27) = 3
7. (a)
1 (b) 1 (c) -1 (d) -1 8. (a) 1 (b) 4 (c) 0 (d) 0
9. (a) -2 (b) -6 (c) x2+4x-2 10. 2
II. (a) odd (b) even (c) odd (d) odd
I, §4, p. 20
1. 8 and 9 2.! and -1 3. /6 and 2 4. and 2
1/3 5. -h and! 6. 9 and 8
7. -! and -1 8.! and! 9. 1 and -! 10. -5:2 and!
A2 ANSWERS TO EXERCISES
II. Yes. Suppose a is negative, so write a = -b where b is positive. Let c be a positive number such that c" = b. Then (-c)" = a because (-1)" = -1 since n is odd. II, §1, p. 24
3. x negative, y positive 4. x negative, y negative
II, §3, p. 33
5. y = + i 6. y = -ix + S 7. x = J2
9 9j3 8. y = --;:;-x + 4 ---;:;-9. y = 4x -3 10. y = -2x + 2
",3+3 ",3+3 II. y= -!x+3+ J'!-12. y=j3x+S+j3 19. -! 20.-8
21. 2 + j2 22.!<3 + j3) 23. y = (x -n>(J22_ n) + 1
24. 25. y= -(x+ l)(j23+ 1)+2
26. y = (x + 1)(3 + j2) + j2 29. (a) x = -4, y = -7 (b) x = y = i
(c) x = -!, y = 1 (d) x = -6, Y = -5 II, §4, p. 35
1.J97 2.j2 3.Js2 4.Ji3 5.!j5 6.(4,-3) 7. SandS 8.(-2,5)
9. Sand 7
II, §8, p. 51
5. (x -2)2 + (y + 1)2 = 25 6. x
2 + (y -
1)2 = 9 7. (x + \)2 + y2 = 3
8. y + ¥ = 2(x + W 9. y - 1 = (x + 2)2 10. y + 4 = (x -\)2
II. (x + 1)2 + (y -2)2 = 2 12. (x -2)2 + (y -\)2 = 2 13. x + ¥-= 2(y + !)2 14. x-I = (y + 2)2
III, §1, p. 61
1. 4 2. -2 3. 2 4. i S. -! 6. 0 7. 4 8. 6 9. 3 10. 12 II. 2
12. 3 13. a
ANSWERS TO EXERCISES
III, §2, p. 70
Tangent line at x = 2 Slope at x = 2
1. 2x y = 4x - 3 4
2. 3x2 y = 12x -16 12
3. 6x2 y = 24x -32 24
4.6x y = 12x -12 12
5.2x y = 4x - 9 4
6. 4x + I Y = 9x - 8 9
7. 4x -3 Y = 5x - 8 5
3x2 8. 2+2 y = 8x - 8 8
9. Y = + 1 -(x + 1)2 -9 2 Y = +.!j
2 10. - (x + 1)2 -9 III, §3, p. 75
2 I 1 I. 4x + 3 2. -2 3. 2 4. 2x + 1 5. -2 6. 9x
2 (2x + 1) (x + 1) (2x -1) 3x 2 10. 2 + I 11. _2/X2 12. -3/x
2 13. -2/(2x -W 14. -3/(3x + 1)2 15. -1/(x + 5)2 16. -I/(x _ 2)2
17. -2x-3 18. -2(x + 1)-3
III, §4, p. 78
1. X4 + 4x
3h + 6x 2 h 2 + 4xh3 + h4 2. 4x3 3. (a) jx-
I/3 (b) (c) 4. Y = 9x -8 5. Y = tx + t slope t -3 7 -3 1 -fi 1 6. Y = Y x + 32' slope y 7. Y = 2-fi x + 2' slope 2-fi
8. (a) !5-
3/4 (b) _!r
- log x derivative formula
- log x derivative by first principle
- log x derivative proof
- log x dérivée
- log x differentiation
- log x^2 derivative
- 1/log x derivative
- log x^3 derivative
17. -2 < x < 8 18. 2 < x < 4 19. -4 < x < 10 20. x < -4 and x> 10
21. x < -10 and x > 4
I, §3, p. 17
1.2. (2x 1) 3. 0, 2, 108 4. 2z -Z2, 2w - w
25. x # j2 or -j2. f(5) = b 6. All x. f(27) = 3
7. (a)
1 (b) 1 (c) -1 (d) -1 8. (a) 1 (b) 4 (c) 0 (d) 0
9. (a) -2 (b) -6 (c) x2+4x-2 10. 2
II. (a) odd (b) even (c) odd (d) odd
I, §4, p. 20
1. 8 and 9 2.! and -1 3. /6 and 2 4. and 2
1/35. -h and! 6. 9 and 8
7. -! and -1 8.! and! 9. 1 and -! 10. -5:2 and!
A2 ANSWERS TO EXERCISES
II. Yes. Suppose a is negative, so write a = -b where b is positive. Let c be a positive number such that c" = b. Then (-c)" = a because (-1)" = -1 since n is odd.II, §1, p. 24
3. x negative, y positive 4. x negative, y negative
II, §3, p. 33
5. y = + i 6. y = -ix + S 7. x = J2
9 9j38. y = --;:;-x + 4 ---;:;-9. y = 4x -3 10. y = -2x + 2
",3+3 ",3+3II. y= -!x+3+ J'!-12. y=j3x+S+j3 19. -! 20.-8
21. 2 + j2 22.!<3 + j3) 23. y = (x -n>(J22_ n) + 1
24. 25. y= -(x+ l)(j23+ 1)+2
26. y = (x + 1)(3 + j2) + j2 29. (a) x = -4, y = -7 (b) x = y = i
(c) x = -!, y = 1 (d) x = -6, Y = -5II, §4, p. 35
1.J97 2.j2 3.Js2 4.Ji3 5.!j5 6.(4,-3) 7. SandS 8.(-2,5)
9. Sand 7
II, §8, p. 51
5. (x -2)2 + (y + 1)2 = 25 6. x
2 + (y -
1)2 = 9 7. (x + \)2 + y2 = 3
8. y + ¥ = 2(x + W 9. y - 1 = (x + 2)2 10. y + 4 = (x -\)2
II. (x + 1)2 + (y -2)2 = 2 12. (x -2)2 + (y -\)2 = 213. x + ¥-= 2(y + !)2 14. x-I = (y + 2)2
III, §1, p. 61
1. 4 2. -2 3. 2 4. i S. -! 6. 0 7. 4 8. 6 9. 3 10. 12 II. 2
12. 3 13. a
ANSWERS TO EXERCISES
III, §2, p. 70
Tangent line at x = 2 Slope at x = 2
1. 2x y = 4x - 3 4
2. 3x2 y = 12x -16 12
3. 6x2 y = 24x -32 24
4.6x y = 12x -12 12
5.2x y = 4x - 9 4
6. 4x + I Y = 9x - 8 9
7. 4x -3 Y = 5x - 8 5
3x28. 2+2 y = 8x - 8 8
9. Y = + 1 -(x + 1)2 -9 2Y = +.!j
2 10. - (x + 1)2 -9III, §3, p. 75
2 I 1I. 4x + 3 2. -2 3. 2 4. 2x + 1 5. -2 6. 9x
2 (2x + 1) (x + 1) (2x -1) 3x 210. 2 + I 11. _2/X2 12. -3/x
213. -2/(2x -W 14. -3/(3x + 1)2 15. -1/(x + 5)2 16. -I/(x _ 2)2
17. -2x-3 18. -2(x + 1)-3
III, §4, p. 78
1. X4 + 4x
3h + 6x 2 h 2 + 4xh3 + h4 2. 4x33. (a) jx-
I/3 (b) (c) 4. Y = 9x -8 5. Y = tx + t slope t -3 7 -3 1 -fi 16. Y = Y x + 32' slope y 7. Y = 2-fi x + 2' slope 2-fi
8. (a) !5-
3/4 (b) _!rAnswers to Exercises
I am much indebted to Anthony Petrello for some of the answers to the exer cises.I, §2, p. 13
1. -3 < x < 3 2. - 1 x
0 3. -j3 x -1 or 1 x j3
4. x < 3 or x > 7 5. - 1 < x < 2 6. x < -lor
x > 1 7. - 5 < x < 58. - 1 x 0 9. x 1 or x = 0 10. x -10 or x = 5
11. x -10 or x = 5 12. x 1 or x = -! 13. x < -414. -5 17. -2 < x < 8 18. 2 < x < 4 19. -4 < x < 10 20. x < -4 and x> 10
21. x < -10 and x > 4
I, §3, p. 17
1. 2. (2x 1) 3. 0, 2, 108 4. 2z -Z2, 2w - w
2 5. x # j2 or -j2. f(5) = b 6. All x. f(27) = 3
7. (a)
1 (b) 1 (c) -1 (d) -1 8. (a) 1 (b) 4 (c) 0 (d) 0
9. (a) -2 (b) -6 (c) x2+4x-2 10. 2
II. (a) odd (b) even (c) odd (d) odd
I, §4, p. 20
1. 8 and 9 2.! and -1 3. /6 and 2 4. and 2
1/3 5. -h and! 6. 9 and 8
7. -! and -1 8.! and! 9. 1 and -! 10. -5:2 and!
A2 ANSWERS TO EXERCISES
II. Yes. Suppose a is negative, so write a = -b where b is positive. Let c be a positive number such that c" = b. Then (-c)" = a because (-1)" = -1 since n is odd. II, §1, p. 24
3. x negative, y positive 4. x negative, y negative
II, §3, p. 33
5. y = + i 6. y = -ix + S 7. x = J2
9 9j3 8. y = --;:;-x + 4 ---;:;-9. y = 4x -3 10. y = -2x + 2
",3+3 ",3+3 II. y= -!x+3+ J'!-12. y=j3x+S+j3 19. -! 20.-8
21. 2 + j2 22.!<3 + j3) 23. y = (x -n>(J22_ n) + 1
24. 25. y= -(x+ l)(j23+ 1)+2
26. y = (x + 1)(3 + j2) + j2 29. (a) x = -4, y = -7 (b) x = y = i
(c) x = -!, y = 1 (d) x = -6, Y = -5 II, §4, p. 35
1.J97 2.j2 3.Js2 4.Ji3 5.!j5 6.(4,-3) 7. SandS 8.(-2,5)
9. Sand 7
II, §8, p. 51
5. (x -2)2 + (y + 1)2 = 25 6. x
2 + (y -
1)2 = 9 7. (x + \)2 + y2 = 3
8. y + ¥ = 2(x + W 9. y - 1 = (x + 2)2 10. y + 4 = (x -\)2
II. (x + 1)2 + (y -2)2 = 2 12. (x -2)2 + (y -\)2 = 2 13. x + ¥-= 2(y + !)2 14. x-I = (y + 2)2
III, §1, p. 61
1. 4 2. -2 3. 2 4. i S. -! 6. 0 7. 4 8. 6 9. 3 10. 12 II. 2
12. 3 13. a
ANSWERS TO EXERCISES
III, §2, p. 70
Tangent line at x = 2 Slope at x = 2
1. 2x y = 4x - 3 4
2. 3x2 y = 12x -16 12
3. 6x2 y = 24x -32 24
4.6x y = 12x -12 12
5.2x y = 4x - 9 4
6. 4x + I Y = 9x - 8 9
7. 4x -3 Y = 5x - 8 5
3x2 8. 2+2 y = 8x - 8 8
9. Y = + 1 -(x + 1)2 -9 2 Y = +.!j
2 10. - (x + 1)2 -9 III, §3, p. 75
2 I 1 I. 4x + 3 2. -2 3. 2 4. 2x + 1 5. -2 6. 9x
2 (2x + 1) (x + 1) (2x -1) 3x 2 10. 2 + I 11. _2/X2 12. -3/x
2 13. -2/(2x -W 14. -3/(3x + 1)2 15. -1/(x + 5)2 16. -I/(x _ 2)2
17. -2x-3 18. -2(x + 1)-3
III, §4, p. 78
1. X4 + 4x
3h + 6x 2 h 2 + 4xh3 + h4 2. 4x3 3. (a) jx-
I/3 (b) (c) 4. Y = 9x -8 5. Y = tx + t slope t -3 7 -3 1 -fi 1 6. Y = Y x + 32' slope y 7. Y = 2-fi x + 2' slope 2-fi
8. (a) !5-
3/4 (b) _!r
- log x derivative formula
- log x derivative by first principle
- log x derivative proof
- log x dérivée
- log x differentiation
- log x^2 derivative
- 1/log x derivative
- log x^3 derivative
17. -2 < x < 8 18. 2 < x < 4 19. -4 < x < 10 20. x < -4 and x> 10
21. x < -10 and x > 4
I, §3, p. 17
1.2. (2x 1) 3. 0, 2, 108 4. 2z -Z2, 2w - w
25. x # j2 or -j2. f(5) = b 6. All x. f(27) = 3
7. (a)
1 (b) 1 (c) -1 (d) -1 8. (a) 1 (b) 4 (c) 0 (d) 0
9. (a) -2 (b) -6 (c) x2+4x-2 10. 2
II. (a) odd (b) even (c) odd (d) odd
I, §4, p. 20
1. 8 and 9 2.! and -1 3. /6 and 2 4. and 2
1/35. -h and! 6. 9 and 8
7. -! and -1 8.! and! 9. 1 and -! 10. -5:2 and!
A2 ANSWERS TO EXERCISES
II. Yes. Suppose a is negative, so write a = -b where b is positive. Let c be a positive number such that c" = b. Then (-c)" = a because (-1)" = -1 since n is odd.II, §1, p. 24
3. x negative, y positive 4. x negative, y negative
II, §3, p. 33
5. y = + i 6. y = -ix + S 7. x = J2
9 9j38. y = --;:;-x + 4 ---;:;-9. y = 4x -3 10. y = -2x + 2
",3+3 ",3+3II. y= -!x+3+ J'!-12. y=j3x+S+j3 19. -! 20.-8
21. 2 + j2 22.!<3 + j3) 23. y = (x -n>(J22_ n) + 1
24. 25. y= -(x+ l)(j23+ 1)+2
26. y = (x + 1)(3 + j2) + j2 29. (a) x = -4, y = -7 (b) x = y = i
(c) x = -!, y = 1 (d) x = -6, Y = -5II, §4, p. 35
1.J97 2.j2 3.Js2 4.Ji3 5.!j5 6.(4,-3) 7. SandS 8.(-2,5)
9. Sand 7
II, §8, p. 51
5. (x -2)2 + (y + 1)2 = 25 6. x
2 + (y -
1)2 = 9 7. (x + \)2 + y2 = 3
8. y + ¥ = 2(x + W 9. y - 1 = (x + 2)2 10. y + 4 = (x -\)2
II. (x + 1)2 + (y -2)2 = 2 12. (x -2)2 + (y -\)2 = 213. x + ¥-= 2(y + !)2 14. x-I = (y + 2)2
III, §1, p. 61
1. 4 2. -2 3. 2 4. i S. -! 6. 0 7. 4 8. 6 9. 3 10. 12 II. 2
12. 3 13. a
ANSWERS TO EXERCISES
III, §2, p. 70
Tangent line at x = 2 Slope at x = 2
1. 2x y = 4x - 3 4
2. 3x2 y = 12x -16 12
3. 6x2 y = 24x -32 24
4.6x y = 12x -12 12
5.2x y = 4x - 9 4
6. 4x + I Y = 9x - 8 9
7. 4x -3 Y = 5x - 8 5
3x28. 2+2 y = 8x - 8 8
9. Y = + 1 -(x + 1)2 -9 2Y = +.!j
2 10. - (x + 1)2 -9III, §3, p. 75
2 I 1I. 4x + 3 2. -2 3. 2 4. 2x + 1 5. -2 6. 9x
2 (2x + 1) (x + 1) (2x -1) 3x 210. 2 + I 11. _2/X2 12. -3/x
213. -2/(2x -W 14. -3/(3x + 1)2 15. -1/(x + 5)2 16. -I/(x _ 2)2
17. -2x-3 18. -2(x + 1)-3
III, §4, p. 78
1. X4 + 4x
3h + 6x 2 h 2 + 4xh3 + h4 2. 4x33. (a) jx-
I/3 (b) (c) 4. Y = 9x -8 5. Y = tx + t slope t -3 7 -3 1 -fi 16. Y = Y x + 32' slope y 7. Y = 2-fi x + 2' slope 2-fi
8. (a) !5-
3/4 (b) _!r- log x derivative formula
- log x derivative by first principle
- log x derivative proof
- log x dérivée
- log x differentiation
- log x^2 derivative
- 1/log x derivative
- log x^3 derivative