Derivative of a function MCQs With Answer

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Derivative of a function MCQs With Answer — This concise introduction helps B. Pharm students master the derivative of a function, a core calculus concept used in pharmacokinetics, drug absorption rates, and formulation optimization. Learn differentiation rules (power, product, quotient, chain), interpretation of instantaneous rate of change, and applications to concentration–time curves, clearance, and half-life calculations. These targeted MCQs reinforce analytical skills needed for modeling drug behavior, estimating slopes of dose–response graphs, and solving related-rate problems in pharmaceutical contexts. Clear explanations and practice improve problem-solving for exams and research. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the derivative of f(x) = x^3 with respect to x?

  • 3x^2
  • x^2
  • 2x
  • 3x

Correct Answer: 3x^2

Q2. The derivative represents which of the following in a concentration–time profile?

  • Instantaneous rate of change of drug concentration
  • Total drug amount absorbed
  • Area under the curve
  • Maximum concentration only

Correct Answer: Instantaneous rate of change of drug concentration

Q3. What is d/dx (e^{2x})?

  • 2e^{2x}
  • e^{2x}
  • 2xe^{2x}
  • e^{x}

Correct Answer: 2e^{2x}

Q4. If C(t) = 50e^{-0.2t}, what does C'(t) represent?

  • Rate of elimination at time t
  • Total dose administered
  • Peak concentration value only
  • Half-life value

Correct Answer: Rate of elimination at time t

Q5. The power rule: d/dx(x^n) equals which expression?

  • n x^{n-1}
  • x^{n+1}/n
  • n x^{n}
  • x^{n-1}/n

Correct Answer: n x^{n-1}

Q6. What is the derivative of ln x (natural log) for x > 0?

  • 1/x
  • ln x
  • x
  • e^{x}

Correct Answer: 1/x

Q7. Using the product rule, d/dx [u(x)v(x)] equals:

  • u’v + uv’
  • u’v’ + uv
  • u’v’ – uv
  • uv

Correct Answer: u’v + uv’

Q8. For v(t) = 5t^2 – 3t + 2, the instantaneous rate dv/dt at t = 2 is:

  • 17
  • 10
  • 19
  • 7

Correct Answer: 17

Q9. The derivative d/dx(sin x) equals:

  • cos x
  • -sin x
  • sin x
  • -cos x

Correct Answer: cos x

Q10. If plasma concentration C(t) has a maximum at t0, then C'(t0) is:

  • Zero
  • Positive and large
  • Undefined only
  • Equal to C(t0)

Correct Answer: Zero

Q11. Chain rule: d/dx f(g(x)) equals:

  • f'(g(x)) g'(x)
  • f(g(x)) + g(x)
  • f'(x) g(x)
  • g'(f(x))

Correct Answer: f'(g(x)) g'(x)

Q12. Differentiate y = (3x + 2)^4. The derivative y’ is:

  • 4(3x + 2)^3 * 3
  • 4(3x + 2)^3
  • (3x + 2)^4 * 3
  • 12x(3x + 2)^3

Correct Answer: 4(3x + 2)^3 * 3

Q13. The derivative of cos x is:

  • -sin x
  • sin x
  • cos x
  • -cos x

Correct Answer: -sin x

Q14. For concentration function C(t) = at + b, the derivative C'(t) equals:

  • a
  • b
  • at
  • 0

Correct Answer: a

Q15. The quotient rule for d/dx [u/v] is:

  • (u’v – uv’)/v^2
  • (u’v + uv’)/v
  • (uv’ – u’v)/v^2
  • u’/v’

Correct Answer: (u’v – uv’)/v^2

Q16. If f(x) = x^{-2}, f'(x) equals:

  • -2x^{-3}
  • -x^{-3}
  • 2x^{-1}
  • 2x^{-3}

Correct Answer: -2x^{-3}

Q17. In pharmacokinetics, the slope of ln C versus time for first-order elimination equals:

  • -k (elimination rate constant)
  • k/2
  • k^2
  • Zero

Correct Answer: -k (elimination rate constant)

Q18. Derivative of tan x is:

  • sec^2 x
  • csc^2 x
  • sec x
  • tan x

Correct Answer: sec^2 x

Q19. If y = ln(5x), dy/dx is:

  • 1/x
  • 1/(5x)
  • 5/x
  • ln 5

Correct Answer: 1/x

Q20. The second derivative f”(x) gives information about:

  • Concavity and acceleration of change
  • Only the slope sign
  • Original function value
  • Area under curve

Correct Answer: Concavity and acceleration of change

Q21. Differentiate f(x) = 7. The derivative f'(x) is:

  • 0
  • 7
  • x
  • 1

Correct Answer: 0

Q22. If concentration C(t) satisfies dC/dt = -0.1 C, what type of elimination is this?

  • First-order elimination
  • Zero-order elimination
  • Mixed-order elimination
  • Michaelis-Menten only

Correct Answer: First-order elimination

Q23. The derivative of 1/x^2 is:

  • -2/x^3
  • 2/x^3
  • -1/x
  • 1/x^2

Correct Answer: -2/x^3

Q24. Implicit differentiation is used when:

  • y is defined implicitly by an equation with x and y mixed
  • Function is linear only
  • There is no y variable
  • Only for polynomials

Correct Answer: y is defined implicitly by an equation with x and y mixed

Q25. For f(x) = x sin x, f'(x) equals:

  • sin x + x cos x
  • x cos x
  • cos x – x sin x
  • sin x

Correct Answer: sin x + x cos x

Q26. Which derivative rule is most appropriate for y = (ln x)^3?

  • Chain rule
  • Quotient rule
  • Product rule only
  • Power of a constant rule

Correct Answer: Chain rule

Q27. The derivative of arctan x is:

  • 1/(1+x^2)
  • 1/(1-x^2)
  • arctan x
  • x/(1+x^2)

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

Q28. If A(t) = 100/(1+t), A'(t) equals:

  • -100/(1+t)^2
  • 100/(1+t)^2
  • -100/(1+t)
  • 0

Correct Answer: -100/(1+t)^2

Q29. In related rates, dC/dt relates to which concept?

  • How concentration changes as time changes
  • Only the initial concentration
  • Total drug administered
  • Half-life as a constant

Correct Answer: How concentration changes as time changes

Q30. Differentiating f(x) = x^2 ln x requires:

  • Product rule and ln derivative
  • Only power rule
  • Only chain rule
  • Implicit differentiation

Correct Answer: Product rule and ln derivative

Q31. If drug concentration decreases linearly, C(t) = C0 – kt, what is the elimination order?

  • Zero-order elimination
  • First-order elimination
  • Second-order elimination
  • Mixed-order elimination

Correct Answer: Zero-order elimination

Q32. The derivative of sinh x is:

  • cosh x
  • sinh x
  • sech x
  • tanh x

Correct Answer: cosh x

Q33. For concentration C(t) = At^n, the time when slope is maximum depends on:

  • Exponent n and coefficient A
  • Only A
  • Only n if A=1
  • Does not depend on n or A

Correct Answer: Exponent n and coefficient A

Q34. The derivative of e^{x} * sin x requires which rules?

  • Product rule with exponential and trigonometric derivatives
  • Chain rule only
  • Quotient rule
  • Power rule only

Correct Answer: Product rule with exponential and trigonometric derivatives

Q35. If f'(a) > 0 at point a, the function is locally:

  • Increasing at a
  • Decreasing at a
  • Maximum at a
  • Constant at a

Correct Answer: Increasing at a

Q36. Find d/dx (x e^{x}). The derivative is:

  • e^{x}(1 + x)
  • xe^{x}
  • e^{x}
  • x e^{x-1}

Correct Answer: e^{x}(1 + x)

Q37. For f(x) = (x^2 + 1)^{-1}, f'(x) equals:

  • -2x/(x^2 + 1)^2
  • 2x/(x^2 + 1)^2
  • -1/(x^2+1)^2
  • 1/(x^2+1)

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

Q38. When optimizing drug release, setting derivative equal to zero helps find:

  • Extremum points (max or min release rate)
  • Only the minimum release
  • Only the initial release
  • Area under release curve

Correct Answer: Extremum points (max or min release rate)

Q39. d/dx (a^x) where a > 0 (constant) is:

  • a^x ln a
  • a^x
  • x a^{x-1}
  • ln x

Correct Answer: a^x ln a

Q40. If concentration changes according to Michaelis-Menten kinetics at low substrate, derivative approximates:

  • First-order with respect to substrate
  • Zero-order always
  • Second-order only
  • No change

Correct Answer: First-order with respect to substrate

Q41. Which rule applies to differentiate y = (x^2 + 1)(ln x)?

  • Product rule combined with ln derivative
  • Chain rule only
  • Quotient rule
  • Implicit differentiation

Correct Answer: Product rule combined with ln derivative

Q42. The derivative of 3^x at x=0 equals:

  • ln 3
  • 3
  • 0
  • 1

Correct Answer: ln 3

Q43. Which statement about differentiability is true?

  • If a function is differentiable at x, it is continuous at x
  • If a function is continuous at x, it must be differentiable at x
  • Differentiability and continuity are unrelated
  • Continuous functions are never differentiable

Correct Answer: If a function is differentiable at x, it is continuous at x

Q44. For rate of infusion R and volume V, concentration C = R/V. If R changes with time, dC/dt depends on:

  • dR/dt divided by V
  • Only V
  • R times V
  • Integral of R

Correct Answer: dR/dt divided by V

Q45. Differentiate f(x) = ln(sin x) for 0

  • cos x / sin x
  • sin x / cos x
  • ln(cos x)
  • 1/sin x

Correct Answer: cos x / sin x

Q46. The derivative of x^{1/2} is:

  • 1/(2 x^{1/2})
  • 1/2
  • x^{-1/2}
  • 2 x^{1/2}

Correct Answer: 1/(2 x^{1/2})

Q47. If f'(x) changes sign from positive to negative at x0, f has:

  • A local maximum at x0
  • A local minimum at x0
  • An inflection point only
  • No extremum

Correct Answer: A local maximum at x0

Q48. The derivative of arccos x is:

  • -1/√(1-x^2)
  • 1/√(1-x^2)
  • arccos x
  • 1/(1+x^2)

Correct Answer: -1/√(1-x^2)

Q49. For a dose-response curve y = A/(1+e^{-k(x-x0)}), the slope dy/dx is largest near:

  • x ≈ x0 (inflection point)
  • x → ∞ only
  • x → -∞ only
  • Where y = 0

Correct Answer: x ≈ x0 (inflection point)

Q50. Which derivative technique helps when y appears on both sides, e.g., x^2 + y^2 = 1?

  • Implicit differentiation
  • Product rule only
  • Logarithmic differentiation only
  • Separation of variables

Correct Answer: Implicit differentiation

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