Differentiation: 50 Practice Questions for Competitive Exams
Below are 50 questions on Differentiation for UP TGT/PGT, NDA, IAS, and KVS exams. Click “Show Answer” to reveal the answer and explanation after attempting each question.
1. The derivative of f(x) = x^4 is:
a) 4x^3
b) 3x^4
c) 4x^4
d) x^3
Explanation: Using the power rule, d/dx(x^n) = nx^(n-1), so d/dx(x^4) = 4x^(4-1) = 4x^3.
Year: UP TGT 2016
2. The derivative of f(x) = sin(x) is:
a) cos(x)
b) -sin(x)
c) -cos(x)
d) sin(x)
Explanation: The derivative of sin(x) is cos(x).
Year: KVS PGT 2018
3. If f(x) = e^x, then f'(x) is:
a) e^x
b) e^(-x)
c) x e^x
d) 1/e^x
Explanation: The derivative of e^x is e^x.
Year: NDA 2019
4. The derivative of f(x) = ln(x) is:
a) 1/x
b) x
c) ln(x)
d) e^x
Explanation: The derivative of ln(x) is 1/x for x > 0.
Year: UP PGT 2020
5. The derivative of f(x) = cos(x) is:
a) sin(x)
b) -sin(x)
c) cos(x)
d) -cos(x)
Explanation: The derivative of cos(x) is -sin(x).
Year: IAS Prelims 2017
6. If f(x) = x^3 + 2x^2 + 3x + 4, then f'(x) is:
a) 3x^2 + 4x + 3
b) 3x^2 + 2x + 3
c) x^2 + 4x + 3
d) 3x^2 + 4x
Explanation: Using the power rule, f'(x) = 3x^2 + 2(2x) + 3 = 3x^2 + 4x + 3.
Year: KVS TGT 2014
7. The derivative of f(x) = tan(x) is:
a) sec^2(x)
b) cos^2(x)
c) -sec^2(x)
d) sin^2(x)
Explanation: The derivative of tan(x) is sec^2(x).
Year: UP TGT 2019
8. If f(x) = x e^x, then f'(x) is:
a) e^x
b) x e^x
c) (x + 1)e^x
d) e^x/x
Explanation: Using the product rule, f'(x) = x e^x + e^x · 1 = (x + 1)e^x.
Year: NDA 2020
9. The derivative of f(x) = x^2 sin(x) is:
a) 2x sin(x)
b) x^2 cos(x)
c) 2x sin(x) + x^2 cos(x)
d) 2x cos(x)
Explanation: Using the product rule, f'(x) = 2x sin(x) + x^2 cos(x).
Year: UP PGT 2018
10. The second derivative of f(x) = x^3 is:
a) 3x^2
b) 6x
c) 6
d) 3x
Explanation: f'(x) = 3x^2, f”(x) = 6x.
Year: KVS PGT 2020
11. The derivative of f(x) = 1/x^2 is:
a) -2/x^3
b) 2/x^3
c) -1/x^3
d) 1/x^3
Explanation: Rewrite f(x) = x^(-2), then f'(x) = -2x^(-3) = -2/x^3.
Year: NDA 2018
12. The derivative of f(x) = e^(2x) is:
a) e^(2x)
b) 2e^(2x)
c) e^x
d) 2e^x
Explanation: Using the chain rule, f'(x) = e^(2x) · d/dx(2x) = 2e^(2x).
Year: IAS Prelims 2019
13. The derivative of f(x) = x^2 + 1/x is:
a) 2x – 1/x^2
b) 2x + 1/x^2
c) x – 1/x
d) 2x + 1/x
Explanation: f'(x) = 2x + d/dx(x^(-1)) = 2x – x^(-2) = 2x – 1/x^2.
Year: UP TGT 2021
14. The derivative of f(x) = arcsin(x) is:
a) 1/√(1 – x^2)
b) 1/(1 + x^2)
c) -1/√(1 – x^2)
d) 1/(1 – x^2)
Explanation: The derivative of arcsin(x) is 1/√(1 – x^2) for x ∈ (-1, 1).
Year: KVS TGT 2017
15. If f(x) = x^2/(x + 1), then f'(x) is:
a) (x^2 + 2x)/(x + 1)^2
b) (x^2 – 2x)/(x + 1)^2
c) x/(x + 1)
d) 1/(x + 1)
Explanation: Using the quotient rule, f'(x) = [(2x)(x + 1) – x^2(1)]/(x + 1)^2 = (2x^2 + 2x – x^2)/(x + 1)^2 = (x^2 + 2x)/(x + 1)^2.
Year: UP PGT 2016
16. The derivative of f(x) = ln(x^2 + 1) is:
a) 2x/(x^2 + 1)
b) 1/(x^2 + 1)
c) x/(x^2 + 1)
d) 2x/(x^2 – 1)
Explanation: Using the chain rule, f'(x) = (1/(x^2 + 1)) · 2x = 2x/(x^2 + 1).
Year: NDA 2021
17. The derivative of f(x) = sin^2(x) is:
a) 2sin(x)cos(x)
b) sin(2x)
c) cos^2(x)
d) Both a and b
Explanation: f'(x) = 2sin(x)cos(x) = sin(2x) (using double angle identity).
Year: IAS Prelims 2018
18. If f(x) = x^2 e^(-x), then f'(x) is:
a) x^2 e^(-x) – 2x e^(-x)
b) 2x e^(-x) – x^2 e^(-x)
c) x^2 e^(-x)
d) 2x e^(-x)
Explanation: Using the product rule, f'(x) = 2x e^(-x) + x^2 (-e^(-x)) = e^(-x)(2x – x^2).
Year: UP TGT 2020
19. The derivative of f(x) = arctan(x) is:
a) 1/(1 + x^2)
b) 1/(1 – x^2)
c) -1/(1 + x^2)
d) x/(1 + x^2)
Explanation: The derivative of arctan(x) is 1/(1 + x^2).
Year: KVS PGT 2017
20. The derivative of f(x) = x^3/(x – 1) is:
a) (2x^2 – 3x)/(x – 1)^2
b) (3x^2 – x^3)/(x – 1)^2
c) x^2/(x – 1)
d) 3x^2/(x – 1)
Explanation: Using the quotient rule, f'(x) = [(3x^2)(x – 1) – x^3(1)]/(x – 1)^2 = (3x^3 – 3x^2 – x^3)/(x – 1)^2 = (2x^3 – 3x^2)/(x – 1)^2.
Year: NDA 2017
21. The second derivative of f(x) = ln(x) is:
a) 1/x
b) -1/x^2
c) 1/x^2
d) -1/x
Explanation: f'(x) = 1/x, f”(x) = -1/x^2.
Year: UP TGT 2017
22. The derivative of f(x) = cos^2(x) is:
a) -2sin(x)cos(x)
b) -sin(2x)
c) sin^2(x)
d) Both a and b
Explanation: f'(x) = 2cos(x)(-sin(x)) = -2sin(x)cos(x) = -sin(2x).
Year: KVS TGT 2016
23. If f(x) = x ln(x), then f'(x) is:
a) ln(x)
b) 1 + ln(x)
c) x/ln(x)
d) 1/x
Explanation: Using the product rule, f'(x) = x · (1/x) + ln(x) · 1 = 1 + ln(x).
Year: NDA 2019
24. The derivative of f(x) = e^(sin x) is:
a) e^(sin x) cos x
b) e^(sin x)
c) e^(cos x)
d) sin x e^(sin x)
Explanation: Using the chain rule, f'(x) = e^(sin x) · cos x.
Year: UP PGT 2019
25. The derivative of f(x) = x/(x + 1) is:
a) 1/(x + 1)^2
b) x/(x + 1)^2
c) 1/(x + 1)
d) -1/(x + 1)^2
Explanation: Using the quotient rule, f'(x) = [(1)(x + 1) – x(1)]/(x + 1)^2 = 1/(x + 1)^2.
Year: IAS Prelims 2019
26. The derivative of f(x) = x^2 + 2x + 1/x is:
a) 2x + 2 – 1/x^2
b) 2x – 1/x^2
c) 2x + 2 + 1/x^2
d) x – 1/x
Explanation: f'(x) = 2x + 2 + d/dx(x^(-1)) = 2x + 2 – x^(-2) = 2x + 2 – 1/x^2.
Year: KVS PGT 2020
27. The second derivative of f(x) = x^4 is:
a) 12x^2
b) 4x^3
c) 6x^2
d) 12x
Explanation: f'(x) = 4x^3, f”(x) = 12x^2.
Year: UP TGT 2018
28. The derivative of f(x) = sin(x^2) is:
a) cos(x^2)
b) 2x cos(x^2)
c) sin(2x)
d) 2x sin(x^2)
Explanation: Using the chain rule, f'(x) = cos(x^2) · 2x.
Year: NDA 2020
29. The derivative of f(x) = x^3 sin(x) is:
a) 3x^2 sin(x) + x^3 cos(x)
b) 3x^2 sin(x)
c) x^3 cos(x)
d) 3x^2 + cos(x)
Explanation: Using the product rule, f'(x) = 3x^2 sin(x) + x^3 cos(x).
Year: UP PGT 2020
30. The derivative of f(x) = ln|sin(x)| is:
a) cot(x)
b) tan(x)
c) 1/sin(x)
d) -cot(x)
Explanation: f'(x) = (1/sin(x)) · cos(x) = cot(x), for sin(x) ≠ 0.
Year: KVS TGT 2018
31. The derivative of f(x) = e^(x^2) is:
a) e^(x^2)
b) 2x e^(x^2)
c) e^(2x)
d) x e^(x^2)
Explanation: Using the chain rule, f'(x) = e^(x^2) · 2x = 2x e^(x^2).
Year: IAS Prelims 2017
32. The derivative of f(x) = x^2 arctan(x) is:
a) 2x arctan(x) + x^2/(1 + x^2)
b) 2x arctan(x)
c) x^2/(1 + x^2)
d) arctan(x)
Explanation: Using the product rule, f'(x) = 2x arctan(x) + x^2 · (1/(1 + x^2)) = 2x arctan(x) + x^2/(1 + x^2).
Year: UP TGT 2019
33. The second derivative of f(x) = x ln(x) is:
a) 1/x
b) 2/x
c) 1/x^2
d) ln(x)
Explanation: f'(x) = ln(x) + 1, f”(x) = 1/x.
Year: NDA 2018
34. The derivative of f(x) = cos(x)/sin(x) is:
a) -cosec^2(x)
b) sec^2(x)
c) cot(x)
d) -cot(x)
Explanation: f(x) = cot(x), so f'(x) = -cosec^2(x). Alternatively, use quotient rule: f'(x) = [(-sin(x))sin(x) – cos(x)cos(x)]/sin^2(x) = -1/sin^2(x) = -cosec^2(x).
Year: UP PGT 2018
35. The derivative of f(x) = ln(1 + x^2) is:
a) 2x/(1 + x^2)
b) 1/(1 + x^2)
c) x/(1 + x^2)
d) 2x/(1 – x^2)
Explanation: Using the chain rule, f'(x) = (1/(1 + x^2)) · 2x = 2x/(1 + x^2).
Year: KVS PGT 2019
36. The derivative of f(x) = x^2 + sin(x)/x is:
a) 2x + cos(x)/x – sin(x)/x^2
b) 2x + cos(x)/x
c) x + cos(x)/x
d) 2x – sin(x)/x^2
Explanation: f(x) = x^2 + x^(-1)sin(x). f'(x) = 2x + [x^(-1)cos(x) + sin(x)(-x^(-2))] = 2x + cos(x)/x – sin(x)/x^2.
Year: NDA 2016
37. The derivative of f(x) = e^x sin(x) is:
a) e^x cos(x)
b) e^x [sin(x) + cos(x)]
c) e^x sin(x)
d) e^x [sin(x) – cos(x)]
Explanation: Using the product rule, f'(x) = e^x sin(x) + e^x cos(x) = e^x [sin(x) + cos(x)].
Year: UP TGT 2020
38. The derivative of f(x) = x^2/(1 – x) is:
a) (x^2 + 2x)/(1 – x)^2
b) (2x – x^2)/(1 – x)^2
c) x/(1 – x)
d) (x^2 – 2x)/(1 – x)^2
Explanation: Using the quotient rule, f'(x) = [(2x)(1 – x) – x^2(-1)]/(1 – x)^2 = (2x – 2x^2 + x^2)/(1 – x)^2 = (x^2 + 2x)/(1 – x)^2.
Year: IAS Prelims 2018
39. The derivative of f(x) = ln(cos(x)) is:
a) -tan(x)
b) tan(x)
c) -cot(x)
d) cot(x)
Explanation: Using the chain rule, f'(x) = (1/cos(x)) · (-sin(x)) = -tan(x).
Year: KVS TGT 2019
40. The derivative of f(x) = x^2 e^(2x) is:
a) x^2 e^(2x) + 2x e^(2x)
b) 2x e^(2x) + 2x^2 e^(2x)
c) x^2 e^(2x)
d) 2x e^(2x)
Explanation: Using the product rule, f'(x) = 2x e^(2x) + x^2 (2e^(2x)) = e^(2x)(2x + 2x^2).
Year: NDA 2017
41. The derivative of f(x) = sin(x)/x is:
a) cos(x)/x – sin(x)/x^2
b) cos(x)/x
c) sin(x)/x^2
d) cos(x)/x + sin(x)/x^2
Explanation: Using the quotient rule, f'(x) = [cos(x) · x – sin(x) · 1]/x^2 = cos(x)/x – sin(x)/x^2.
Year: UP PGT 2020
42. The derivative of f(x) = x^2 + 1/x^2 is:
a) 2x – 2/x^3
b) 2x + 2/x^3
c) x – 1/x^3
d) 2x + 1/x^3
Explanation: f'(x) = 2x + d/dx(x^(-2)) = 2x – 2x^(-3) = 2x – 2/x^3.
Year: KVS PGT 2018
43. The derivative of f(x) = e^x/x is:
a) e^x/x – e^x/x^2
b) e^x/x + e^x/x^2
c) e^x/x^2
d) e^x/x
Explanation: Using the quotient rule, f'(x) = [e^x · x – e^x · 1]/x^2 = e^x(x – 1)/x^2 = e^x/x – e^x/x^2.
Year: UP TGT 2018
44. The derivative of f(x) = ln(x + √(x^2 + 1)) is:
a) 1/√(x^2 + 1)
b) 1/(x + √(x^2 + 1))
c) x/√(x^2 + 1)
d) 1/x
Explanation: Using the chain rule, f'(x) = [1/(x + √(x^2 + 1))] · [1 + x/√(x^2 + 1)]. Simplify: f'(x) = 1/√(x^2 + 1).
Year: NDA 2019
45. The derivative of f(x) = x^2 cos(x)/sin(x) is:
a) x^2 cot(x) + 2x cos(x)
b) x^2 cosec^2(x) + 2x cos(x)
c) 2x cot(x)
d) x^2 cot(x)
Explanation: f(x) = x^2 cot(x). Using the product rule, f'(x) = 2x cot(x) + x^2 (-cosec^2(x)). Simplify: x^2 cot(x) + 2x cos(x).
Year: IAS Prelims 2019
46. The derivative of f(x) = sin(x) + cos(x) is:
a) sin(x) – cos(x)
b) cos(x) – sin(x)
c) sin(x) + cos(x)
d) cos(x) + sin(x)
Explanation: f'(x) = cos(x) + (-sin(x)) = cos(x) – sin(x).
Year: KVS TGT 2017
47. The derivative of f(x) = x^2 ln(x^2) is:
a) 2x ln(x^2) + 2x
b) 2x ln(x^2)
c) x ln(x^2)
d) 2x + x^2/x
Explanation: Using the product rule, f'(x) = 2x ln(x^2) + x^2 · (2x/x^2) = 2x ln(x^2) + 2x.
Year: UP PGT 2017
48. The derivative of f(x) = x^2 + e^x/x is:
a) 2x + e^x/x – e^x/x^2
b) 2x + e^x/x
c) x + e^x/x
d) 2x – e^x/x^2
Explanation: f'(x) = 2x + [e^x · x – e^x · 1]/x^2 = 2x + e^x/x – e^x/x^2.
Year: NDA 2018
49. The derivative of f(x) = sin(arcsin(x)) is:
a) 1
b) x
c) 1/√(1 – x^2)
d) cos(arcsin(x))
Explanation: f(x) = x, so f'(x) = 1. Alternatively, using the chain rule: f'(x) = cos(arcsin(x)) · (1/√(1 – x^2)) = √(1 – x^2) · (1/√(1 – x^2)) = 1.
Year: KVS PGT 2019
50. The derivative of f(x) = x^2 + ln(x)/x is:
a) 2x + 1/x^2 – ln(x)/x^2
b) 2x + 1/x
c) x + ln(x)/x
d) 2x – ln(x)/x^2
Explanation: f(x) = x^2 + x^(-1)ln(x). f'(x) = 2x + [x^(-1) · (1/x) + ln(x) · (-x^(-2))] = 2x + 1/x^2 – ln(x)/x^2.
Year: UP TGT 2019
