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Chain Rule (No trig or exponential)
1. If y = (2x + 3)⁴, then dy/dx = A) 4(2x + 3)³
B) 4x(2x + 3)³
C) 8(2x + 3)³
D) 4(2x + 3)⁴
2. If y = (x² + 1)⁵, find dy/dx.
A) 10x(x² + 1)⁴
B) 5(x² + 1)⁴
C) 5x(x² + 1)⁴
D) 2x(x² + 1)⁵
3. Differentiate y = √(3x² + 1).
A) 3x / √(3x² + 1)
B) 6x / √(3x² + 1)
C) 3x / (2√(3x² + 1))
D) 6x√(3x² + 1)
4. If y = (5x − 1)³, dy/dx =
A) 3(5x − 1)²
B) 15(5x − 1)²
C) 3(5x − 1)³
D) 15x²
5. Find d/dx of y = (x² + 4x + 4)³.
A) 6x(x + 2)²
B) 3(x² + 4x + 4)²(2x + 4)
C) (x² + 4x + 4)³
D) (2x + 4)³
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Trigonometric Differentiation (sec, cosec, cot only)
6. d/dx (sec x) =
A) sec x tan x
B) sec² x
C) sec x cot x
D) −sec x tan x
7. d/dx (cosec x) =
A) −cosec x cot x
B) −sec x tan x
C) cosec x cot x
D) sec x cot x
8. d/dx (cot x) =
A) −cosec² x
B) −sec² x
C) cosec² x
D) −cot x cosec x
9. Which of the following is the derivative of f(x) = 1 / cot x?
A) csc² x / cot² x
B) −cosec² x / cot² x
C) csc x
D) −cosec² x / cot² x
10. Which function has derivative −cosec x cot x?
A) cot x
B) cosec x
C) sec x
D) tan x
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Implicit Differentiation
11. If x² + y² = 25, find dy/dx.
A) −x / y
B) x / y
C) y / x
D) −y / x
12. If x³ + y³ = 6xy, find dy/dx.
A) (2y − x²) / (y² − 2x)
B) (2x − y²) / (x² − 2y)
C) (2y − x²) / (3y² − 6x)
D) (2x − y²) / (3x² − 6y)
13. Differentiate implicitly: sin x + cos y = 1.
A) dy/dx = −cos x / sin y
B) dy/dx = sin x / sin y
C) dy/dx = −cos x / (−sin y)
D) dy/dx = cos x / cos y
14. If x²y + y³ = x, find dy/dx.
A) (1 − 2xy − y²) / (x² + 3y²)
B) (1 − 2xy) / (x² + y²)
C) (1 − 2xy − 3y²) / (x² + 3y²)
D) (1 − x²) / (2xy + 3y²)
15. Differentiate implicitly: e^y + x = y.
A) dy/dx = 1 / (1 − e^y)
B) dy/dx = (1 − e^y)
C) dy/dx = (e^y − 1)
D) dy/dx = −1 / (1 + e^y)
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Turning Points, Stationary Points, Maxima/Minima (Calculation)
16. Find the stationary point of y = x³ − 3x² + 2.
A) x = 0, 2
B) x = 1
C) x = 1, 2
D) x = 0, 3
17. Determine the nature of the turning point of y = x² − 4x + 4.
A) Maximum
B) Minimum
C) Point of inflection
D) None
18. Find x for which y = x³ − 3x has a maximum.
A) x = −1
B) x = 0
C) x = 1
D) x = −2
19. If y = −x² + 4x, what type of turning point occurs at x = 2?
A) Maximum
B) Minimum
C) Saddle point
D) None
20. At which x does y = x³ − 3x² + 4 have a point of inflection?
A) x = 0
B) x = 1
C) x = 2
D) x = 3
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Increasing/Decreasing Function (Calculation & Concept)
21. f(x) = x³ − 6x² + 9x is increasing when:
A) x > 3
B) x < 0
C) 1 < x < 3
D) x > 0
22. A function is decreasing if:
A) f '(x) > 0
B) f '(x) < 0
C) f ''(x) > 0
D) f ''(x) < 0
23. Find the interval where f(x) = x⁴ − 4x² is decreasing.
A) (−∞, −√2) ∪ (0, √2)
B) (−√2, 0)
C) (√2, ∞)
D) (−∞, −√2) ∪ (√2, ∞)
24. If f '(x) = 0 at x = 2 and f ''(2) > 0, then:
A) Maximum at x = 2
B) Minimum at x = 2
C) Increasing at x = 2
D) Decreasing at x = 2
25. The function f(x) = x³ − 3x² has a local maximum at:
A) x = 0
B) x = 1
C) x = 2
D) x = 3
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Tangents and Normals
26. Find the slope of the tangent to y = x² at x = 3.
A) 3
B) 6
C) 2
D) 9
27. The slope of the normal to y = x² + 1 at x = 1 is:
A) −2
B) −1/2
C) 1/2
D) 2
28. The tangent to y = x³ at x = 2 passes through:
A) (2, 8)
B) (2, 12)
C) (2, 6)
D) (2, 16)
29. The equation of the tangent to y = x² at x = 2 is:
A) y = 4x − 4
B) y = 4x + 4
C) y = 2x + 4
D) y = 2x − 4
30. Normal to y = x² at x = 1 has slope:
A) −2
B) −1/2
C) 1/2
D) 2
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Theoretical Questions (Conceptual, Non-Calculation)
31. A stationary point occurs when:
A) dy/dx = 0
B) dy/dx ≠ 0
C) dy/dx = ∞
D) dy/dx = 1
32. A function is increasing when:
A) f '(x) = 0
B) f '(x) < 0
C) f '(x) > 0
D) f ''(x) > 0
33. At a point of inflection, the second derivative is:
A) > 0
B) < 0
C) = 0
D) undefined
34. If f ''(x) > 0 at a stationary point, the point is a:
A) Maximum
B) Minimum
C) Inflection
D) Saddle point
35. The slope of the normal is the:
A) Same as derivative
B) Negative reciprocal of tangent’s slope
C) Product of slopes is 1
D) None of the above
36. If the derivative of a function is always positive, then the function is:
A) Constant
B) Decreasing
C) Increasing
D) Undefined
37. The chain rule is used when:
A) Two functions are added
B) A function is inside another function
C) Two functions are divided
D) A function is multiplied by x
38. The quotient rule is applied to:
A) A constant function
B) f(x) + g(x)
C) f(x)/g(x)
D) f(g(x))
39. Implicit differentiation is required when:
A) y is explicitly in terms of x
B) y cannot be separated
C) Equation involves only x
D) y is constant
40. At a turning point:
A) dy/dx = 0
B) dy/dx = ∞
C) dy/dx = 1
D) dy/dx ≠ 0
