Derivative of a constant × function MCQs With Answer

Derivative of a constant × function MCQs With Answer

This concise introduction explains the constant multiple rule in differentiation tailored for B.Pharm students. The derivative of a constant times a function equals the constant multiplied by the derivative of the function, a foundational concept in calculus for pharmacy applications. Understanding this rule aids in analyzing drug concentration-time profiles, infusion rates, elimination kinetics, and dose-response models. Practice with varied functions—polynomials, exponentials, logarithms, and trigonometric forms—reinforces applying the rule alongside chain and product rules in pharmacokinetics problems. These MCQs emphasize practical examples and calculations to build fluency for exams and real-world pharmaceutical modeling. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. If f(t) = 5·t^3, what is f'(t)?

• 15·t^2
• 5·t^2
• 3·t^4
• 0

Correct Answer: 15·t^2

Q2. For g(x) = 7·sin(x), what is g'(x)?

• 7·cos(x)
• 7·(-cos(x))
• 7·cos(x)·x’
• sin(x)

Correct Answer: 7·cos(x)

Q3. If h(t) = k·e^{-0.2t} where k is a constant, dh/dt equals:

• -0.2·k·e^{-0.2t}
• -k·e^{-0.2t}
• 0.2·k·e^{-0.2t}
• k·e^{-0.2t}

Correct Answer: -0.2·k·e^{-0.2t}

Q4. The derivative of a constant c (no x) is:

• c’
• 1
• 0
• c

Correct Answer: 0

Q5. If F(x) = 4·ln(x), F'(x) is:

• 4/x
• ln(x)/4
• 1/(4x)
• 4·(1/ln x)

Correct Answer: 4/x

Q6. For dose-response model S(t) = 3·t^2 + 2, what is S'(t)?

• 6·t
• 6·t + 2
• 3·2·t
• 0

Correct Answer: 6·t

Q7. If C(t) = 10·cos(ωt), the rate dC/dt equals:

• -10·ω·sin(ωt)
• 10·ω·sin(ωt)
• -10·sin(t)
• 10·cos(ωt)

Correct Answer: -10·ω·sin(ωt)

Q8. Given M(x) = a·x^5 where a is constant, M”(x) (second derivative) is:

• 20·a·x^3
• 5·a·x^4
• a·x^3
• 0

Correct Answer: 20·a·x^3

Q9. If R(t) = 2·(t^3 + t), R'(t) equals:

• 6·t^2 + 2
• 2·3·t^2 + t
• 3·t^2 + 1
• 6·t^2 + 1

Correct Answer: 6·t^2 + 2

Q10. For function U(x) = 0·x^2, U'(x) is:

• 0
• 0·2x
• 0 for all x
• All of the above

Correct Answer: All of the above

Q11. If Y(t) = k·sin^2(t) and k is constant, Y'(t) is:

• k·2·sin(t)·cos(t)
• 2·k·cos^2(t)
• k·sin(2t)
• Both options 1 and 3 are equivalent

Correct Answer: Both options 1 and 3 are equivalent

Q12. In pharmacokinetics, concentration C = C0·e^{-kt}. The derivative dC/dt is:

• -k·C0·e^{-kt}
• -C0·e^{-kt}
• k·C0·e^{-kt}
• C0·e^{-kt}

Correct Answer: -k·C0·e^{-kt}

Q13. If P(x) = 5·(3x^2 + x), P'(x) equals:

• 5·(6x + 1)
• 30x + 5
• 15x + 5
• Both options 1 and 3

Correct Answer: Both options 1 and 3

Q14. For V(t) = a·t^{-1} where a constant, V'(t) equals:

• -a·t^{-2}
• a·t^{-2}
• -a/t
• 0

Correct Answer: -a·t^{-2}

Q15. The constant multiple rule states d/dx[c·f(x)] =:

• c·f'(x)
• f'(x)/c
• c’·f(x)
• f(x)·c’

Correct Answer: c·f'(x)

Q16. If A(t) = 4·tan(t), A'(t) is:

• 4·sec^2(t)
• 4·tan'(t)·t’
• sec^2(t)
• 4·cos^2(t)

Correct Answer: 4·sec^2(t)

Q17. For S(x) = c·x where c constant, S'(x) equals:

• c
• x
• 1
• 0

Correct Answer: c

Q18. If B(t) = 6·e^{2t}, what is dB/dt?

• 12·e^{2t}
• 6·2·e^{2t}
• 6·e^{2t}
• Both options 1 and 2

Correct Answer: Both options 1 and 2

Q19. When differentiating c·f(x) and c depends on x, the correct rule is:

• c·f'(x)
• c’·f(x) + c·f'(x)
• 0
• f(x)·c

Correct Answer: c’·f(x) + c·f'(x)

Q20. If E(t) = 7·(3t + 1)^2, E'(t) equals:

• 7·2·(3t + 1)·3
• 42·(3t + 1)·t’
• 14·(3t + 1)
• 14·3·(3t + 1)

Correct Answer: 7·2·(3t + 1)·3

Q21. For L(x) = 9·x^{-3}, L'(x) is:

• -27·x^{-4}
• -9·3·x^{-4}
• -27/x^4
• All of the above

Correct Answer: All of the above

Q22. If function f(x) = c·cosh(x) with c constant, f'(x) equals:

• c·sinh(x)
• c·cosh(x)
• sinh(x)
• c’·cosh(x)

Correct Answer: c·sinh(x)

Q23. In drug infusion, rate R(t) = k·t where k constant. Instantaneous change dR/dt is:

• k
• t
• 0
• kt

Correct Answer: k

Q24. If Q(x) = 2·(x^3) + 5, Q”(x) is:

• 12·x
• 6·x
• 12·x + 0
• Both options 1 and 3

Correct Answer: Both options 1 and 3

Q25. For T(t) = c·(sin(t) + cos(t)), T'(t) is:

• c·(cos(t) – sin(t))
• c·(cos(t) + sin(t))
• cos(t) – sin(t)
• c’·(sin(t) + cos(t))

Correct Answer: c·(cos(t) – sin(t))

Q26. If f(x) = 8·(1/x), f'(x) equals:

• -8·x^{-2}
• 8/x
• -1/(8x^2)
• 0

Correct Answer: -8·x^{-2}

Q27. For drug decay D(t) = A·e^{-kt} with constant A and k, d^2D/dt^2 is:

• k^2·A·e^{-kt}
• A·k^2·e^{-kt}
• A·(-k)^2·e^{-kt}
• All options represent the same value

Correct Answer: All options represent the same value

Q28. If H(x) = 3·arctan(x), H'(x) equals:

• 3/(1 + x^2)
• 1/(1 + x^2)
• 3·(1 + x^2)
• arctan'(x)

Correct Answer: 3/(1 + x^2)

Q29. The derivative of 0·f(x) for any differentiable f(x) is:

• 0
• f'(x)
• Undefined
• Depends on f

Correct Answer: 0

Q30. If V(t) = 5·t·e^{t}, which rule simplifies differentiation using a constant multiple?

• Constant multiple rule then product rule
• Only chain rule
• Only quotient rule
• Ignore the constant then integrate

Correct Answer: Constant multiple rule then product rule

Q31. For W(x) = c·x^0, where x^0 = 1, W'(x) equals:

• 0
• c
• 1
• x^0

Correct Answer: 0

Q32. If Z(t) = 4·sinh(t), Z'(t) is:

• 4·cosh(t)
• 4·sinh(t)
• cosh(t)
• 0

Correct Answer: 4·cosh(t)

Q33. When differentiating c·f(x) + d·g(x) with c,d constants, the derivative is:

• c·f'(x) + d·g'(x)
• c’·f(x) + d’·g(x)
• f'(x) + g'(x)
• c·g'(x) + d·f'(x)

Correct Answer: c·f'(x) + d·g'(x)

Q34. For a pharmacokinetic example, if rate R(t) = 2·ln(1 + t), R'(t) equals:

• 2/(1 + t)
• ln(1 + t)/2
• 1/(2(1 + t))
• 2·(1 + t)

Correct Answer: 2/(1 + t)

Q35. If S(x) = 6·e^{x^2}, S'(x) equals:

• 6·2x·e^{x^2}
• 12x·e^{x^2}
• Both options 1 and 2
• 6·e^{x^2}

Correct Answer: Both options 1 and 2

Q36. For f(t) = c·(t^4 – 2t), the correct derivative f'(t) is:

• c·(4t^3 – 2)
• 4c·t^3 – 2c
• Both options 1 and 2
• 4t^3 – 2

Correct Answer: Both options 1 and 2

Q37. If a is constant and y = a·sin(ax), treating a as constant, dy/dx is:

• a·a·cos(ax)
• a·cos(ax)
• 0
• sin(ax)

Correct Answer: a·a·cos(ax)

Q38. For function f(x) = 10·(x ln x), f'(x) equals:

• 10·(ln x + 1)
• 10·x·(1/x)
• ln x + 1
• 10·ln x

Correct Answer: 10·(ln x + 1)

Q39. If derivative operation is linear, what is d/dx[5·f(x) – 3·g(x)]?

• 5·f'(x) – 3·g'(x)
• 5·f(x) – 3·g(x)
• 5’·f(x) – 3’·g(x)
• 0

Correct Answer: 5·f'(x) – 3·g'(x)

Q40. For kinetic model K(t) = c·t·e^{-kt}, where c and k constants, dK/dt requires:

• constant multiple rule and product & chain rules
• only integration
• only chain rule
• no differentiation needed

Correct Answer: constant multiple rule and product & chain rules

Q41. If M(x) = 2·(sin x)/x, derivative M'(x) is best approached by:

• Pull out constant then use quotient rule
• Differentiate numerator only
• Use constant multiple and product rule with x^{-1}
• Both options 1 and 3

Correct Answer: Both options 1 and 3

Q42. For f(x) = 3·(cos x)^2, f'(x) equals:

• 3·2·cos x·(-sin x)
• -6·cos x·sin x
• -3·sin(2x)
• All of the above are equivalent

Correct Answer: All of the above are equivalent

Q43. If c is constant and y = c·(ax^2 + bx + d), what is y’?

• c·(2ax + b)
• 2acx + bc
• Both options 1 and 2
• 2ax + b

Correct Answer: Both options 1 and 2

Q44. For function F(x) = k·(e^{3x} + x), F'(x) is:

• k·(3e^{3x} + 1)
• 3k·e^{3x} + k
• Both options 1 and 2
• k·e^{3x}

Correct Answer: Both options 1 and 2

Q45. If P(t) = 0·e^{t} + 4·t, P'(t) equals:

• 4
• 0·e^{t} + 4
• 4 + 0
• All of the above

Correct Answer: All of the above

Q46. When differentiating c·f(g(x)) where c constant, you should:

• Take c·[f'(g(x))·g'(x)]
• Differentiate c as variable
• Ignore g'(x)
• Multiply f and g first

Correct Answer: Take c·[f'(g(x))·g'(x)]

Q47. If S(t) = 7·(t^2·e^t), S'(t) equals:

• 7·(2t·e^t + t^2·e^t)
• 7·e^t·(2t + t^2)
• Both options 1 and 2
• 7·(t^2·e^t)

Correct Answer: Both options 1 and 2

Q48. For f(x) = c·(1 + x)^{-2}, f'(x) equals:

• -2c·(1 + x)^{-3}
• -2·(1 + x)^{-3}
• c·(1 + x)^{-1}
• 0

Correct Answer: -2c·(1 + x)^{-3}

Q49. If you see 5·d/dx[sin x], the correct simplification is:

• 5·cos x
• cos x
• 5·sin x
• sin x

Correct Answer: 5·cos x

Q50. In summary, applying the constant multiple rule helps B.Pharm students:

• Simplify derivatives in PK/PD modeling
• Compute rates of change of concentration and dose
• Combine with product and chain rules for complex models
• All of the above

Correct Answer: All of the above

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