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Determine whether each of the following statements is true or false, and explain why.

1. The derivative of \(x^x\) is \(\pi x^{n-1}\).

2. The derivative of a sum is the sum of the derivatives.

3. The derivative of a product is the product of the derivatives.

4. The derivative of a quotient is the quotient of the derivatives.

5. The chain rule is used to take the derivative of a product of functions.

6. The only function that is its own derivative is \(e^x\).

7. The derivative of \(10^x\) is \(\times 10^{x-1}\).

8. The derivative of \(\ln |x|\) is the same as the derivative of \(\ln x\).

9. The derivative of \(\ln kx\) is the same as the derivative of \(\ln x\).

10. The derivative of \(\log x\) is the same as the derivative of \(\ln x\).

11. \(D, \tan(x^2) = \sec^2(x^2)\).

12. \(D, \tan^2 x = 2\tan x \sec^2 x\).

Use the rules for derivatives to find the derivative of each function defined as follows:

13. \(y = 5x^3 - 7x^2 - 9x + 5\).

14. \(y = x^2 + 13x^5 - 4x^4 + 9x^3 - 6x + 4\).

15. \(y = 9x^{mn}\).

16. \(y = -4x^{-1}\).

17. \(f(x) = 3x^{-1} + 6x\).

18. \(f(x) = 19x^{-1} - 8x\).

19. \(L(x) = \frac{4x+7}{3x}\).

20. \(y = 3x^2 + 2x + x - 1 + x - 2\).

21. \(y = \frac{x+2}{2x^3} - 5x^2\).

22. \(f(x) = (3x^2 - 2)^4\).

23. \(y = x + \frac{5}{x^2}\).

24. \(y = \frac{2f}{7} - 5\).

25. \(y = \frac{3}{x^2} + 4x^3\).

26. \(y = 3x(2x+1)^3\).

27. \(r(t) = \frac{(3t+1)^3}{5t^2 - 7t}\).

28. \(y = x^x + \frac{x^2}{x}\).

29. \(p(t) = t^2 (t^2 + 1)^{3/2}\).

30. \(y = x^{40} + 4\sec x - 3\log x\).

31. \(y = -6e^{2x}\).

32. \(y = \sin^3 x + \cos^3 x\).

33. \(y = e^{-2n}\).

34. \(y = 3\sin^2(\cos 4x)\).

35. \(y = 5xe^{2x}\).

36. \(y = \left(x + \frac{x^2}{3}\right)^7\).

37. \(y = \ln(2 + x^2)\).

38. \(y = \frac{9 - 4x}{x}\).

39. \(y = \frac{x - 3}{\ln|3x|}\).

40. \(y = \frac{x}{x \cdot x}\).

41. \(y = \ln(x^2 - 1) \cdot xe^2\).

42. \(y = 2\sin(\sin(\sin 3x))\).

43. \(s = (t^2 + e^2)^2\).

44. \(y = \cos x + 2\cos x\).

45. \(y = 3 \cdot 10^{-x^2}\).

46. \(q = (e^{2p+1} - 2)^4\).

47. \(g(z) = \log_2(z^3 + z + 1)\).

48. \(h(z) = \log(1 + e)\).

49. \(f(x) = e^{2x} \ln(xe^x + 1)\).

50. \(f(x) = \ln(x + 1) e^x\).

51. \(y = 2\tan 5x\).

52. \(y = -4\sin 7x\).

53. \(y = \cot(6 - 3x^2)\).

54. \(y = \sin x\).

55. \(f(x) = 3x^{-1} + 6x\).

56. \(y = \frac{\sin x}{x}\).

57. \(y = 2\sin^4(4x^2)\).

58. \(y = x \sec x\).

59. \(y = \cos(1 + x^2)\).

60. \(y = \cot\left(\frac{21}{x^4}\right)\).

61. \(y = e^{-2x} \sin x\).

62. \(y = \tan^2(x^3)\).

63. \(y = \frac{1 - \cos x}{\cos^2 x}\).

64. \(y = \frac{1 - \sin 2x}{1 + \sin 2x}\).

65. \(y = \frac{1 + x}{\tan x}\).

66. \(y = \frac{\sec x}{6 - x}\).

67. \(y = \ln|5\sin x|\).

68. \(y = \ln|\cos x|\).

Answer :


21. y=3x^2 +2x+x^(-1) +x^(-2)
False. This statement is false because the given equation contains negative exponents (x^(-1) and x^(-2)), which are not allowed in this context. In a polynomial equation, the exponents must be non-negative integers.

22. y=(x+2)/(2x^3 -5x^2)
True. This statement is true because the given equation is a rational function, which is the ratio of two polynomials. The derivative of a rational function can be found using the quotient rule.

23. f(x)=(3x^2 -2)^4
True. This statement is true because the given equation is a composition of a power function (3x^2 -2) raised to the fourth power. The derivative of a power function can be found using the chain rule.

24. y=x+(5/x^2)
True. This statement is true because the given equation is a sum of two terms: x and (5/x^2). The derivative of a sum is the sum of the derivatives.

25. y=(2f^7 -5)^2
False. This statement is false because it is not clear what "f" represents in the equation. Without knowing the specific definition of "f," we cannot determine the derivative of the equation.

26. y=(3/x^2) +4x^3
True. This statement is true because the given equation is a sum of two terms: (3/x^2) and 4x^3. The derivative of a sum is the sum of the derivatives.

27. y=3x(2x+1)^3
True. This statement is true because the given equation is a product of two terms: 3x and (2x+1)^3. The derivative of a product can be found using the product rule.

29. r(t)=(3t+1)^3/(5t^2 -7t)
True. This statement is true because the given equation is a quotient of two terms: (3t+1)^3 and (5t^2 -7t). The derivative of a quotient can be found using the quotient rule.

31. p(t)=t^2 (t^2 +1)^(3/2)
True. This statement is true because the given equation is a composition of a power function (t^2 +1) raised to the 3/2 power. The derivative of a power function can be found using the chain rule.

33. y=-6e^(2x)
True. This statement is true because the given equation is a constant (-6) multiplied by the exponential function e^(2x). The derivative of an exponential function can be found using the chain rule.

35. y=e^(-2n)
True. This statement is true because the given equation is an exponential function e^(-2n) raised to a negative power. The derivative of an exponential function can be found using the chain rule.

37. y=5xe^(2x)
True. This statement is true because the given equation is a product of two terms: 5x and e^(2x). The derivative of a product can be found using the product rule.

38. y=(9-4x)/(x)
True. This statement is true because the given equation is a quotient of two terms: (9-4x) and (x). The derivative of a quotient can be found using the quotient rule.

39. y=ln(2+x^2)
True. This statement is true because the given equation is the natural logarithm of (2+x^2). The derivative of the natural logarithm function can be found using the chain rule.

41. y=(x-3)/ln∣3x∣
True. This statement is true because the given equation is a quotient of two terms: (x-3) and ln∣3x∣. The derivative of a quotient can be found using the quotient rule.

43. y=ln(x^2 -1)/(xe^2)
True. This statement is true because the given equation is a quotient of two terms: ln(x^2 -1) and (xe^2). The derivative of a quotient can be found using the quotient rule.

45. s=(t^2 +e^2)^2
True. This statement is true because the given equation is a composition of a power function (t^2 +e^2) raised to the second power. The derivative of a power function can be found using the chain rule.

46. y=cos(x) +2cos(x)
True. This statement is true because the given equation is a sum of two terms: cos(x) and 2cos(x). The derivative of a sum is the sum of the derivatives.

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