JEE Main - Derivative rules for composite and implicit functions Online Test (MCQ Quiz)

1. What is the derivative of f(x) = sin(x^2) with respect to x?

A) cos(x^2) B) 2x cos(x^2) C) -2x sin(x^2) D) sin(2x)

2. If y = (3x + 2)^4, find dy/dx.

A) 4(3x + 2)^3 B) 12(3x + 2)^3 C) 3(3x + 2)^3 D) (3x + 2)^5 / 5

3. Given f(x) = e^(tan(x)), what is f'(x)?

A) e^(tan(x)) sec^2(x) B) e^(tan(x)) tan(x) C) sec^2(x) e^(tan(x)-1) D) e^(sec^2(x))

4. What is the derivative of g(x) = ln(cos(x))?

A) tan(x) B) -tan(x) C) cot(x) D) -cot(x)

5. If y = sqrt(1 - x^2), find dy/dx.

A) 1 / (2 * sqrt(1 - x^2)) B) -x / sqrt(1 - x^2) C) x / sqrt(1 - x^2) D) -1 / sqrt(1 - x^2)

6. Find the derivative of h(x) = cos(sin(x)).

A) -sin(sin(x)) cos(x) B) sin(sin(x)) cos(x) C) -cos(cos(x)) sin(x) D) sin(cos(x)) sin(x)

7. If y = x^x, what is dy/dx?

A) x^x (1 + ln(x)) B) x^x (ln(x)) C) x * x^(x-1) D) x^x

8. Differentiate f(x) = a^(g(x)) where g(x) is a function of x.

A) a^(g(x)) g'(x) ln(a) B) a^(g(x)) g'(x) C) g'(x) a^(g(x)-1) D) a^(g'(x)) ln(a)

9. What is the derivative of y = arctan(x/a)?

A) 1 / (1 + (x/a)^2) B) a / (a^2 + x^2) C) 1 / (a^2 + x^2) D) 1 / sqrt(1 - (x/a)^2)

10. If x^2 + y^2 = 25, find dy/dx.

A) -x/y B) x/y C) -y/x D) y/x

11. Find the derivative of y = sin(x)cos(x).

A) cos^2(x) - sin^2(x) B) 2cos^2(x) - 1 C) 1 - 2sin^2(x) D) cos(2x)

12. If y = tan(3x), what is dy/dx?

A) sec^2(3x) B) 3 sec^2(3x) C) tan(3x) D) 3 tan(3x)

13. What is the derivative of f(x) = arcsin(2x)?

A) 2 / sqrt(1 - 4x^2) B) 1 / sqrt(1 - 4x^2) C) 2 / sqrt(1 - x^2) D) 1 / sqrt(1 - x^2)

14. Given y = e^(kx), find dy/dx.

A) k e^(kx) B) e^(kx) C) k e^(x) D) e^(kx) / k

15. If y = log_a(x), find dy/dx.

A) 1 / (x ln(a)) B) 1 / x C) ln(a) / x D) 1 / (x log_a(e))

16. Find the derivative of y = x^n * e^x.

A) nx^(n-1)e^x + x^n e^x B) x^n e^x + x^(n-1) e^x C) n x^n e^x D) x^n e^x

17. If y = sqrt(ax + b), find dy/dx.

A) a / (2 * sqrt(ax + b)) B) 1 / (2 * sqrt(ax + b)) C) a * sqrt(ax + b) D) b / (2 * sqrt(ax + b))

18. What is the derivative of f(x) = sec(x^3)?

A) 3x^2 sec(x^3) tan(x^3) B) sec(x^3) tan(x^3) C) 3x^2 sec(x^3) D) sec(3x^2) tan(3x^2)

19. If x^3 + y^3 = 3xy, find dy/dx.

A) (y - x^2) / (y^2 - x) B) (x^2 - y) / (y^2 - x) C) (3y - 3x^2) / (3y^2 - 3x) D) (y^2 - x) / (x^2 - y)

20. Find the derivative of y = sin(ax + b).

A) a cos(ax + b) B) cos(ax + b) C) -a sin(ax + b) D) a sin(ax + b)

21. What is the derivative of f(x) = cot(x^2)?

A) -2x csc^2(x^2) B) csc^2(x^2) C) -csc^2(x^2) D) 2x cot(x^2)

22. If y = e^(sin(x)), find dy/dx.

A) e^(sin(x)) cos(x) B) e^(sin(x)) C) cos(x) e^(cos(x)) D) sin(x) e^(sin(x))

23. Find the derivative of y = x^2 * sin(x).

A) 2x sin(x) + x^2 cos(x) B) 2x cos(x) + x^2 sin(x) C) x^2 cos(x) D) 2x sin(x)

24. What is the derivative of f(x) = log(1 + x^2) with respect to x?

A) 2x / (1 + x^2) B) 1 / (1 + x^2) C) 2x log(1 + x^2) D) log(2x)

25. If y = (ln(x))^2, find dy/dx.

A) 2 ln(x) / x B) ln(x) / x C) 2 ln(x) D) 1 / x

26. Find the derivative of y = sin^{-1}(e^x).

A) e^x / sqrt(1 - e^(2x)) B) 1 / sqrt(1 - e^(2x)) C) e^x / sqrt(1 - e^x) D) e^x cos(e^x)

27. If y = x * arctan(x), find dy/dx.

A) arctan(x) + x / (1 + x^2) B) 1 / (1 + x^2) C) arctan(x) D) x / (1 + x^2)

28. What is the derivative of f(x) = tan^{-1}(x^2)?

A) 2x / (1 + x^4) B) 1 / (1 + x^4) C) 2x sec^2(x^2) D) x^2 / (1 + x^4)

29. If y = x^n, find dy/dx.

A) nx^(n-1) B) n x^n C) x^(n-1) D) n x

30. Find the derivative of y = e^(x^2 + x).

A) (2x + 1) e^(x^2 + x) B) x e^(x^2 + x) C) 2x e^(x^2 + x) D) e^(2x + 1)

31. If y = cos(x^2 + 1), find dy/dx.

A) -2x sin(x^2 + 1) B) 2x sin(x^2 + 1) C) -sin(x^2 + 1) D) -2x cos(x^2 + 1)

32. What is the derivative of f(x) = sqrt(sin(x))?

A) cos(x) / (2 * sqrt(sin(x))) B) sin(x) / (2 * sqrt(cos(x))) C) sqrt(cos(x)) D) 1 / (2 * sqrt(sin(x)))

33. If y = x^3 * e^(2x), find dy/dx.

A) 3x^2 e^(2x) + 2x^3 e^(2x) B) 3x^2 e^(2x) C) 2x^3 e^(2x) D) x^3 e^(2x) + 2x^2 e^(2x)

34. Find the derivative of y = ln(sin(x)).

A) cot(x) B) tan(x) C) -cot(x) D) sec(x)

35. What is the derivative of f(x) = arcsin(x/a)?

A) 1 / sqrt(a^2 - x^2) B) a / sqrt(a^2 - x^2) C) 1 / sqrt(1 - x^2) D) 1 / (a^2 + x^2)

36. If y = e^(tan^{-1}(x)), find dy/dx.

A) e^(tan^{-1}(x)) / (1 + x^2) B) e^(tan^{-1}(x)) C) 1 / (1 + x^2) D) tan^{-1}(x) e^(tan^{-1}(x)-1)

37. Find the derivative of y = x^2 * cos(x).

A) 2x cos(x) - x^2 sin(x) B) 2x cos(x) + x^2 sin(x) C) -x^2 sin(x) D) 2x cos(x)

38. What is the derivative of f(x) = (x^2 + 1)^3?

A) 3(x^2 + 1)^2 * 2x B) (x^2 + 1)^2 * 2x C) 3(x^2 + 1)^2 D) 6x(x^2 + 1)^2

39. If y = sin(x^2) + cos(x^2), find dy/dx.

A) 2x cos(x^2) - 2x sin(x^2) B) cos(x^2) - sin(x^2) C) 2x cos(x^2) + 2x sin(x^2) D) sin(x^2) - cos(x^2)

40. Find the derivative of y = e^(x^3).

A) 3x^2 e^(x^3) B) x^2 e^(x^3) C) e^(x^3) D) 3x^2

41. What is the derivative of f(x) = ln(tan(x))?

A) sec(x) csc(x) B) sec(x) / tan(x) C) csc(x) / tan(x) D) sec^2(x)

42. If y = x^2 * ln(x), find dy/dx.

A) 2x ln(x) + x B) x ln(x) + x^2 C) 2x ln(x) D) x + ln(x)

43. Find the derivative of y = (sin(x))^n.

A) n (sin(x))^(n-1) cos(x) B) n (sin(x))^(n-1) C) n cos(x) (sin(x))^(n-1) D) (sin(x))^n cos(x)

44. What is the derivative of f(x) = tan(e^x)?

A) sec^2(e^x) * e^x B) sec^2(e^x) C) e^x tan(e^x) D) tan(e^x)

45. If y = sqrt(x^2 + a^2), find dy/dx.

A) x / sqrt(x^2 + a^2) B) a / sqrt(x^2 + a^2) C) x / (2 * sqrt(x^2 + a^2)) D) 1 / (2 * sqrt(x^2 + a^2))

46. Find the derivative of y = x * sin^{-1}(x).

A) sin^{-1}(x) + x / sqrt(1 - x^2) B) 1 / sqrt(1 - x^2) C) x / sqrt(1 - x^2) D) sin^{-1}(x)

47. What is the derivative of f(x) = cos(e^x)?

A) -e^x sin(e^x) B) -sin(e^x) C) e^x cos(e^x) D) -e^x cos(e^x)

48. If y = x^2 * e^(sin(x)), find dy/dx.

A) 2x e^(sin(x)) + x^2 e^(sin(x)) cos(x) B) 2x e^(sin(x)) C) x^2 e^(sin(x)) cos(x) D) e^(sin(x)) + x^2 e^(sin(x)) cos(x)

49. Find the derivative of y = ln(x^2 + 1).

A) 2x / (x^2 + 1) B) 1 / (x^2 + 1) C) 2x ln(x^2 + 1) D) 1 / x

50. If y = tan(x^2 + x), find dy/dx.

A) sec^2(x^2 + x) * (2x + 1) B) sec^2(x^2 + x) C) (2x + 1) tan(x^2 + x) D) tan(2x + 1)

Derivative rules for composite and implicit functions - Online Test

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