Differentiability: 50 Practice Questions for Competitive Exams
Below are 50 questions on Differentiability for UP TGT/PGT, NDA, IAS, and KVS exams. Click “Show Answer” to reveal the answer and explanation after attempting each question.
1. The function f(x) = |x| is differentiable at:
a) x = 0
b) x ≠ 0
c) All real numbers
d) No real numbers
Explanation: f(x) = |x| has a sharp corner at x = 0, so it is not differentiable at x = 0. For x ≠ 0, f(x) is linear (x or -x), hence differentiable.
Year: UP TGT 2016
2. If f(x) = x² + 3x + 2, then f'(1) is:
a) 3
b) 4
c) 5
d) 6
Explanation: f(x) = x² + 3x + 2, so f'(x) = 2x + 3. At x = 1, f'(1) = 2(1) + 3 = 5.
Year: KVS PGT 2018
3. The function f(x) = sin(x) is differentiable:
a) Only at x = nπ
b) For all x ∈ R
c) For x ≠ nπ
d) Nowhere
Explanation: sin(x) is a smooth function with derivative cos(x), defined for all real numbers.
Year: NDA 2019
4. If f(x) = e^x, then f”(x) is:
a) e^x
b) 2e^x
c) e^(-x)
d) 0
Explanation: f(x) = e^x, so f'(x) = e^x, and f”(x) = e^x.
Year: UP PGT 2020
5. The function f(x) = [x] (greatest integer function) is differentiable at:
a) All integers
b) All non-integers
c) All real numbers
d) Nowhere
Explanation: [x] is constant between integers, so differentiable (derivative = 0) for non-integers. At integers, it has jumps, so not differentiable.
Year: IAS Prelims 2017
6. If f(x) = x³ – 2x + 1, then f'(2) is:
a) 8
b) 10
c) 12
d) 14
Explanation: f(x) = x³ – 2x + 1, so f'(x) = 3x² – 2. At x = 2, f'(2) = 3(2²) – 2 = 12 – 2 = 10.
Year: KVS TGT 2014
7. The function f(x) = |x – 2| is differentiable at:
a) x = 2
b) x ≠ 2
c) All x ∈ R
d) No x ∈ R
Explanation: f(x) = |x – 2| has a corner at x = 2, so it is not differentiable there. For x ≠ 2, it is linear, hence differentiable.
Year: UP TGT 2019
8. If f(x) = ln(x), then f'(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: NDA 2020
9. The function f(x) = x|x| is differentiable at:
a) x = 0
b) x ≠ 0
c) All x ∈ R
d) No x ∈ R
Explanation: f(x) = x|x| = x² for x ≥ 0 and -x² for x < 0. At x = 0, left derivative = right derivative = 0, and elsewhere, f'(x) = ±2x, so differentiable everywhere.
Year: UP PGT 2018
10. If f(x) = cos(x), then f”(x) is:
a) sin(x)
b) -sin(x)
c) -cos(x)
d) cos(x)
Explanation: f(x) = cos(x), f'(x) = -sin(x), f”(x) = -cos(x).
Year: KVS PGT 2020
11. If f(x) = x² sin(1/x) for x ≠ 0 and f(0) = 0, is f differentiable at x = 0?
a) Yes
b) No
c) Partially
d) Cannot be determined
Explanation: Check f'(0) using the limit definition: lim(h→0) [f(h) – f(0)]/h = lim(h→0) h sin(1/h) = 0 (since |sin(1/h)| ≤ 1). Thus, f is differentiable at x = 0.
Year: NDA 2018
12. The derivative of f(x) = tan(x) at x = π/4 is:
a) 1
b) 2
c) √2
d) 1/√2
Explanation: f'(x) = sec²(x). At x = π/4, sec(π/4) = √2, so f'(π/4) = (√2)² = 2.
Year: IAS Prelims 2019
13. If f(x) = √x, then f'(4) is:
a) 1/4
b) 1/2
c) 1/√4
d) 2
Explanation: f(x) = x^(1/2), so f'(x) = (1/2)x^(-1/2). At x = 4, f'(4) = (1/2)(4^(-1/2)) = 1/(2√4) = 1/4.
Year: UP TGT 2021
14. The function f(x) = 1/x is differentiable for:
a) x = 0
b) x ≠ 0
c) All x ∈ R
d) x > 0
Explanation: f(x) = 1/x has derivative f'(x) = -1/x², defined for x ≠ 0. At x = 0, f(x) is undefined.
Year: KVS TGT 2017
15. If f(x) = x³ + 2x² + 3x + 4, then f”(x) is:
a) 6x + 4
b) 3x² + 4x
c) 6x + 2
d) 3x + 2
Explanation: f'(x) = 3x² + 4x + 3, f”(x) = 6x + 4.
Year: UP PGT 2016
16. The function f(x) = |x² – 1| is differentiable at:
a) x = ±1
b) x ≠ ±1
c) All x ∈ R
d) No x ∈ R
Explanation: f(x) = |x² – 1| has corners at x² – 1 = 0, i.e., x = ±1, where it is not differentiable. Elsewhere, it is differentiable.
Year: NDA 2021
17. If f(x) = e^(2x), then f'(x) 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 2018
18. The derivative of f(x) = sin²(x) is:
a) 2sin(x)cos(x)
b) sin(2x)
c) cos²(x)
d) Both a and b
Explanation: f(x) = sin²(x), f'(x) = 2sin(x)cos(x) = sin(2x) (using double angle identity).
Year: UP TGT 2020
19. 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: KVS PGT 2017
20. The function f(x) = x^(1/3) is differentiable at:
a) x = 0
b) x ≠ 0
c) All x ∈ R
d) No x ∈ R
Explanation: f'(x) = (1/3)x^(-2/3) = 1/(3x^(2/3)), undefined at x = 0 due to vertical tangent.
Year: NDA 2017
21. If f(x) = x² + 1/x, then f'(x) is:
a) 2x – 1/x²
b) 2x + 1/x²
c) x – 1/x
d) 2x – 1/x
Explanation: f'(x) = 2x + d/dx(1/x) = 2x – 1/x².
Year: UP TGT 2017
22. The function f(x) = |sin(x)| is differentiable at:
a) x = nπ
b) x ≠ nπ
c) All x ∈ R
d) No x ∈ R
Explanation: |sin(x)| has corners at sin(x) = 0, i.e., x = nπ, where it is not differentiable. Elsewhere, it is differentiable.
Year: KVS TGT 2016
23. If f(x) = e^(sin x), then f'(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: NDA 2019
24. The second derivative of f(x) = x⁴ is:
a) 12x²
b) 4x³
c) 6x²
d) 12x
Explanation: f'(x) = 4x³, f”(x) = 12x².
Year: UP PGT 2019
25. If f(x) = x/(x + 1), then f'(x) 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 function f(x) = x^(2/3) is differentiable at:
a) x = 0
b) x ≠ 0
c) All x ∈ R
d) No x ∈ R
Explanation: f'(x) = (2/3)x^(-1/3) = 2/(3x^(1/3)), undefined at x = 0 due to vertical tangent.
Year: KVS PGT 2020
27. If f(x) = sin(x) + cos(x), then f'(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: UP TGT 2018
28. The derivative of f(x) = arcsin(x) is:
a) 1/√(1 – x²)
b) 1/(1 + x²)
c) -1/√(1 – x²)
d) 1/(1 – x²)
Explanation: The derivative of arcsin(x) is 1/√(1 – x²) for x ∈ (-1, 1).
Year: NDA 2020
29. 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: UP PGT 2020
30. The function f(x) = |x – 3| + |x + 3| is differentiable at:
a) x = 3
b) x = -3
c) x ≠ ±3
d) All x ∈ R
Explanation: The function has corners at x = 3 and x = -3, where the absolute value expressions change behavior, so not differentiable there.
Year: KVS TGT 2018
31. If f(x) = x²/(x + 1), then f'(x) is:
a) (x² + 2x)/(x + 1)²
b) (x² – 2x)/(x + 1)²
c) x/(x + 1)
d) 1/(x + 1)
Explanation: Using the quotient rule, f'(x) = [(2x)(x + 1) – x²(1)]/(x + 1)² = (2x² + 2x – x²)/(x + 1)² = (x² + 2x)/(x + 1)².
Year: IAS Prelims 2017
32. The second derivative of f(x) = ln(1 + x) is:
a) 1/(1 + x)
b) -1/(1 + x)²
c) 1/(1 + x)²
d) -1/(1 + x)
Explanation: f'(x) = 1/(1 + x), f”(x) = -1/(1 + x)².
Year: UP TGT 2019
33. If f(x) = tan⁻¹(x), then f'(x) is:
a) 1/(1 + x²)
b) 1/(1 – x²)
c) -1/(1 + x²)
d) x/(1 + x²)
Explanation: The derivative of tan⁻¹(x) is 1/(1 + x²).
Year: NDA 2018
34. If f(x) = x sin(x), then f'(x) is:
a) sin(x) + x cos(x)
b) sin(x) – x cos(x)
c) x sin(x)
d) cos(x)
Explanation: Using the product rule, f'(x) = x cos(x) + sin(x) · 1 = sin(x) + x cos(x).
Year: UP PGT 2018
35. The function f(x) = |x² – 4| is differentiable at:
a) x = ±2
b) x ≠ ±2
c) All x ∈ R
d) No x ∈ R
Explanation: f(x) = |x² – 4| has corners at x² – 4 = 0, i.e., x = ±2, where it is not differentiable.
Year: KVS PGT 2019
36. If f(x) = e^(x²), then f'(x) is:
a) e^(x²)
b) 2x e^(x²)
c) e^(2x)
d) x e^(x²)
Explanation: Using the chain rule, f'(x) = e^(x²) · 2x = 2x e^(x²).
Year: NDA 2016
37. The derivative of f(x) = cos²(x) is:
a) -2sin(x)cos(x)
b) -sin(2x)
c) sin²(x)
d) Both a and b
Explanation: f'(x) = 2cos(x)(-sin(x)) = -2sin(x)cos(x) = -sin(2x).
Year: UP TGT 2020
38. If f(x) = ln(x² + 1), then f'(x) is:
a) 2x/(x² + 1)
b) 1/(x² + 1)
c) x/(x² + 1)
d) 2x/(x² – 1)
Explanation: Using the chain rule, f'(x) = (1/(x² + 1)) · 2x = 2x/(x² + 1).
Year: IAS Prelims 2018
39. The function f(x) = x^(1/5) is differentiable at:
a) x = 0
b) x ≠ 0
c) All x ∈ R
d) No x ∈ R
Explanation: f'(x) = (1/5)x^(-4/5), undefined at x = 0 due to vertical tangent.
Year: KVS TGT 2019
40. If f(x) = x³ sin(x), then f'(x) is:
a) 3x² sin(x) + x³ cos(x)
b) 3x² sin(x)
c) x³ cos(x)
d) 3x² + cos(x)
Explanation: Using the product rule, f'(x) = 3x² sin(x) + x³ cos(x).
Year: NDA 2017
41. The second derivative of f(x) = x ln(x) is:
a) 1/x
b) 2/x
c) 1/x²
d) ln(x)
Explanation: f'(x) = ln(x) + 1, f”(x) = 1/x.
Year: UP PGT 2020
42. If f(x) = arctan(x²), then f'(x) is:
a) 2x/(1 + x⁴)
b) x/(1 + x²)
c) 1/(1 + x²)
d) 2x/(1 + x²)
Explanation: Using the chain rule, f'(x) = (1/(1 + (x²)²)) · 2x = 2x/(1 + x⁴).
Year: KVS PGT 2018
43. The function f(x) = |x| + |x – 1| is differentiable at:
a) x = 0
b) x = 1
c) x ≠ 0, 1
d) All x ∈ R
Explanation: The function has corners at x = 0 and x = 1, where the absolute value terms change, so not differentiable there.
Year: UP TGT 2018
44. If f(x) = x² e^(-x), then f'(x) is:
a) x² e^(-x) – 2x e^(-x)
b) 2x e^(-x) – x² e^(-x)
c) x² e^(-x)
d) 2x e^(-x)
Explanation: Using the product rule, f'(x) = 2x e^(-x) + x² (-e^(-x)) = e^(-x)(2x – x²).
Year: NDA 2019
45. If f(x) = ln|sin(x)|, then f'(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: IAS Prelims 2019
46. The derivative of f(x) = x³/(x – 1) is:
a) (2x² – 3x)/(x – 1)²
b) (3x² – x³)/(x – 1)²
c) x²/(x – 1)
d) 3x²/(x – 1)
Explanation: Using the quotient rule, f'(x) = [(3x²)(x – 1) – x³(1)]/(x – 1)² = (3x³ – 3x² – x³)/(x – 1)² = (2x³ – 3x²)/(x – 1)².
Year: KVS TGT 2017
47. The function f(x) = |x² – x| is differentiable at:
a) x = 0, 1
b) x ≠ 0, 1
c) All x ∈ R
d) No x ∈ R
Explanation: f(x) = |x² – x| = |x(x – 1)| has corners at x = 0, 1, where it is not differentiable.
Year: UP PGT 2017
48. If f(x) = sin(x²), then f'(x) is:
a) cos(x²)
b) 2x cos(x²)
c) sin(2x)
d) 2x sin(x²)
Explanation: Using the chain rule, f'(x) = cos(x²) · 2x.
Year: NDA 2018
49. If f(x) = x + 1/x, then f”(x) is:
a) 2/x³
b) -2/x³
c) 1/x²
d) -1/x²
Explanation: f'(x) = 1 – 1/x², f”(x) = 0 + 2/x³ = 2/x³.
Year: KVS PGT 2019
50. The function f(x) = |x – 1| + |x + 1| is differentiable at:
a) x = ±1
b) x ≠ ±1
c) All x ∈ R
d) No x ∈ R
Explanation: The function has corners at x = 1 and x = -1, where the absolute value terms change, so not differentiable there.
Year: UP TGT 2019
