Calculus

Question 1

1. Derivative of (x^3) is:
   A. x² B. 2x² C. 3x² D. 4x²

Answer 1

✅ Answer: C – 3x²
Power rule → multiply exponent by coefficient, subtract 1 from exponent.

Question 2

2. Derivative of (5x^2 + 4x + 3) is:
   A. 10x + 4 B. 5x + 4 C. 10x + 3 D. 9x + 4

Answer 2

✅ Answer: A – 10x + 4
Differentiate each term individually.

Question 3

3. If (y = \sin x), then (dy/dx) is:
   A. cos x B. –cos x C. sin x D. –sin x

Answer 3

✅ Answer: A – cos x
Derivative of sine is cosine.

Question 4

4. If (y = \cos x), then (dy/dx) is:
   A. cos x B. sin x C. –sin x D. –cos x

Answer 4

✅ Answer: C – –sin x
Derivative of cosine is negative sine.

Question 5

5. If (y = e^x), the derivative is:
   A. e B. x e^(x–1) C. e^x D. 1

Answer 5

✅ Answer: C – e^x
Exponential functions remain unchanged after differentiation.

Question 6

6. Derivative of (ln x) is:
   A. 1/x B. x C. ln x D. e^x

Answer 6

✅ Answer: A – 1/x
Natural log derivative rule.

Question 7

7. If (y = \tan x), then (dy/dx =)?
   A. sec x B. sec²x C. cos²x D. sin x

Answer 7

✅ Answer: B – sec²x
Derivative of tangent is secant squared.

Question 8

8. The derivative of a constant is:
   A. Constant B. 1 C. 0 D. Undefined

Answer 8

✅ Answer: C – 0
Constant has no rate of change.

Question 9

9. If (y = x^5 – 3x^3 + 2x), find (dy/dx):
   A. 5x⁴ – 9x² + 2 B. 3x⁴ – 6x² + 2 C. 5x⁴ – 6x² + 2x D. 5x⁴ – 9x² + 1

Answer 9

✅ Answer: A – 5x⁴ – 9x² + 2
Apply power rule on each term.

Question 10

10. Find (dy/dx) if (y = x³ \sin x):
    A. 3x² sin x B. 3x² sin x + x³ cos x C. x³ cos x – 3x² sin x D. sin x + cos x

Answer 10

✅ Answer: B – 3x² sin x + x³ cos x
Use product rule: u′v + uv′.

Question 11

11. If (y = e^{2x}), then (dy/dx =):
    A. e^{2x} B. 2e^{2x} C. 2x e^x D. 2x e^{x²}

Answer 11

✅ Answer: B – 2e^{2x}
Apply chain rule (derivative of inner term 2x).

Question 12

12. If (y = \sin(3x)), then (dy/dx =):
    A. cos(3x) B. 3cos(3x) C. 3sin(3x) D. –sin(3x)

Answer 12

✅ Answer: B – 3cos(3x)
Chain rule → multiply by inner derivative (3).

Question 13

13. If (y = 1/x), derivative is:
    A. 1/x² B. –1/x² C. ln x D. 2x

Answer 13

✅ Answer: B – –1/x²
Rewrite as x⁻¹ → derivative is –1·x⁻².

Question 14

14. The slope of y = x² at x = 2 is:
    A. 2 B. 4 C. 6 D. 8

Answer 14

✅ Answer: B – 4
dy/dx = 2x → substitute x = 2.

Question 15

15. Second derivative of y = x³ is:
    A. 3x² B. 6x C. 9x D. 12x

Answer 15

✅ Answer: B – 6x
First derivative 3x² → derivative again = 6x.

Question 16

16. Evaluate ∫x dx.
    A. x² + C B. x²/2 + C C. 2x + C D. x + C

Answer 16

✅ Answer: B – x²/2 + C
Power rule for integration.

Question 17

17. ∫sin x dx =
    A. cos x + C B. –cos x + C C. tan x + C D. –tan x + C

Answer 17

✅ Answer: B – –cos x + C
Integral of sine is –cosine.

Question 18

18. ∫e^x dx =
    A. e^x + C B. e^(x²) + C C. xe^x + C D. ln e + C

Answer 18

✅ Answer: A – e^x + C
Exponential integrates to itself.

Question 19

19. ∫1/x dx =
    A. x + C B. e^x + C C. ln|x| + C D. 1/x + C

Answer 19

✅ Answer: C – ln|x| + C
Reciprocal rule → log of absolute value.

Question 20

20. ∫x² dx =
    A. x³/3 + C B. 2x + C C. 3x² + C D. x²/2 + C

Answer 20

✅ Answer: A – x³/3 + C
Add 1 to exponent, divide by new exponent.

Question 21

21. If dy/dx = 6x – 4, find y.
    A. 6x² – 4x + C B. 3x² – 4x + C C. 3x² – 2x + C D. 2x³ – 4x + C

Answer 21

✅ Answer: B – 3x² – 4x + C
Integrate each term.

Question 22

22. Derivative of (x² e^x) is:
    A. 2x e^x + x² e^x B. e^x + 2x² C. 2x e^x D. x e^x

Answer 22

✅ Answer: A – 2x e^x + x² e^x
Product rule again.

Question 23

23. If (y = ln(2x)), then dy/dx =
    A. 2/x B. 1/x C. 1/2x D. 2x

Answer 23

✅ Answer: B – 1/x
Derivative = (1/2x)·2 → simplifies to 1/x.

Question 24

24. ∫cos x dx =
    A. –sin x + C B. sin x + C C. tan x + C D. –tan x + C

Answer 24

✅ Answer: B – sin x + C
Integral of cosine is sine.

Question 25

25. The area under y = x from 0 to 2 is: A. 1 B. 2 C. 3 D. 4

Answer 25

✅ Answer: B – 2 ∫₀² x dx = [x²/2]₀² = 2.