Derivative of a function MCQs With Answer

Derivative of a function MCQs With Answer offers B. Pharm students a focused review of derivatives as tools for analysing drug concentration, rate of change, and optimization in pharmaceutical processes. This concise, keyword-rich introduction covers core concepts such as limits, power rule, product and quotient rules, chain rule, implicit differentiation, higher-order derivatives, and their applications in pharmacokinetics (absorption, distribution, elimination). Emphasis is placed on interpreting slopes, critical points, concavity, and linear approximation to solve real-world formulation and dosage problems. Each MCQ reinforces analytical skills essential for modeling rates and optimizing drug therapy. 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
• 2x
• x^2
• 3x

Correct Answer: 3x^2

Q2. If concentration C(t) = 5e^{-0.2t} mg/L, what is dC/dt at time t?

• -1.0e^{-0.2t}
• -0.2·5e^{-0.2t}
• 0.2·5e^{-0.2t}
• 5e^{-0.2t}

Correct Answer: -0.2·5e^{-0.2t}

Q3. Which rule is used to differentiate h(x) = (x^2)(sin x)?

• Quotient rule
• Chain rule
• Product rule
• Power rule only

Correct Answer: Product rule

Q4. The derivative of ln x is:

• 1/x
• ln(x)/x
• 1/ln x
• x

Correct Answer: 1/x

Q5. For y = (3x + 2)^4, dy/dx is found using:

• Product rule
• Chain rule
• Quotient rule
• Implicit differentiation

Correct Answer: Chain rule

Q6. If rate of drug absorption A(t) = t^2, what is the instantaneous rate of change at t = 3?

• 3
• 6
• 9
• 18

Correct Answer: 6

Q7. d/dx (e^{3x}) equals:

• 3e^{3x}
• e^{3x}/3
• e^{x}
• 3x e^{3x}

Correct Answer: 3e^{3x}

Q8. If f'(x) = 0 at x = a, then x = a is necessarily:

• A local maximum
• A local minimum
• A critical point
• Inflection point

Correct Answer: A critical point

Q9. The quotient rule for d/dx [u/v] is:

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

Correct Answer: (u’v – uv’)/v^2

Q10. Second derivative f”(x) provides information about:

• Function value
• Slope only
• Concavity and inflection points
• Limits at infinity

Correct Answer: Concavity and inflection points

Q11. Differentiate y = x^{1/2}. dy/dx =

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

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

Q12. For implicit function defined by x^2 + y^2 = 25, dy/dx at (3,4) is:

• -x/y = -3/4
• x/y = 3/4
• -y/x = -4/3
• y/x = 4/3

Correct Answer: -x/y = -3/4

Q13. If drug amount A(t) has derivative A'(t) < 0, this means:

• Drug amount is increasing
• Drug amount is decreasing
• Drug amount is constant
• Slope is positive

Correct Answer: Drug amount is decreasing

Q14. d/dx (tan x) equals:

• sec x
• sec^2 x
• csc^2 x
• tan x

Correct Answer: sec^2 x

Q15. Which derivative rule applies to y = ln(sin x)?

• Product rule
• Chain rule
• Quotient rule
• Power rule

Correct Answer: Chain rule

Q16. The derivative of 1/x is:

• -1/x^2
• 1/x^2
• -x
• 0

Correct Answer: -1/x^2

Q17. If f(x) = x^4 – 4x^3, critical points are found by solving:

• f(x) = 0
• f'(x) = 0
• f”(x) = 0
• ∫f(x) dx = 0

Correct Answer: f'(x) = 0

Q18. d/dx [arctan x] equals:

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

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

Q19. For concentration profile C(t) = t/(1+t), dC/dt at t=1 is:

• 1/4
• 1/2
• 1/3
• 0

Correct Answer: 1/4

Q20. The derivative of f(x) = x^n (n constant) is:

• nx^{n-1}
• x^{n+1}/n
• n^x
• nx^n

Correct Answer: nx^{n-1}

Q21. If velocity v(t) is derivative of displacement s(t), then acceleration a(t) is:

• The integral of v(t)
• The derivative of v(t)
• Equal to s(t)
• Zero

Correct Answer: The derivative of v(t)

Q22. Use derivative tests: if f'(x) changes from + to – at x=a, x=a is:

• An inflection point
• A local minimum
• A local maximum
• No special point

Correct Answer: A local maximum

Q23. Differentiate f(x) = e^{x^2}. f'(x) =

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

Correct Answer: 2x e^{x^2}

Q24. For y = x^2 sin x, derivative at x=0 is:

• 0
• 1
• 2
• Undefined

Correct Answer: 0

Q25. The derivative of cos x is:

• sin x
• -sin x
• -cos x
• cos x

Correct Answer: -sin x

Q26. d/dx [log_a x] equals:

• 1/(x ln a)
• ln a / x
• 1/x
• ln x / a

Correct Answer: 1/(x ln a)

Q27. If concentration C has maximum at t = t0, then C'(t0) equals:

• Positive
• Negative
• Zero
• Undefined

Correct Answer: Zero

Q28. The derivative of x sin x using product rule is:

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

Correct Answer: sin x + x cos x

Q29. For function f(x)=x^5-5x^3, f”(x) helps determine:

• Concavity and inflection
• Zero-crossings only
• Limits at infinity
• First derivative values

Correct Answer: Concavity and inflection

Q30. If y = x^x, dy/dx equals:

• x^x(ln x + 1)
• x^{x-1}
• x^x ln x
• x/\ln x

Correct Answer: x^x(ln x + 1)

Q31. Which derivative represents sensitivity (elasticity) of response R to dose D if R(D)=D^k?

• kD^{k-1}
• D^k
• kD
• ln(D)/k

Correct Answer: kD^{k-1}

Q32. d/dx [sec x] equals:

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

Correct Answer: sec x tan x

Q33. For rate law R = kC^2, dR/dC =

• 2kC
• kC^2
• k/(2C)
• k

Correct Answer: 2kC

Q34. If f(x) = sin^2 x, which rule is best to differentiate?

• Product rule then chain rule
• Only power rule
• Only quotient rule
• No rule needed

Correct Answer: Product rule then chain rule

Q35. The derivative of arccos x is:

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

Correct Answer: -1/√(1-x^2)

Q36. Linear approximation of f at x=a uses f'(a) to estimate f(a+h) as:

• f(a) + f'(a)h
• f'(a) + f(a)h
• f(a) – f'(a)h
• f(a)h

Correct Answer: f(a) + f'(a)h

Q37. For C(t)=At + B, second derivative C”(t) equals:

• 0
• A
• B
• At+B

Correct Answer: 0

Q38. When differentiating implicitly y^3+x^3=6xy, dy/dx is found using:

• Separate variables
• Implicit differentiation and algebra
• Integration
• Logarithmic differentiation only

Correct Answer: Implicit differentiation and algebra

Q39. d/dx [x/(1+x)] simplified equals:

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

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

Q40. If concentration curve changes concavity at t0, then f”(t0) is:

• Zero or undefined (possible inflection)
• Always positive
• Always negative
• Equal to f'(t0)

Correct Answer: Zero or undefined (possible inflection)

Q41. The derivative of sinh x is:

• cosh x
• sinh x
• tanh x
• 1

Correct Answer: cosh x

Q42. For f(x)=ln(x^2+1), f'(x) equals:

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

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

Q43. If drug concentration follows C(t)=At e^{-kt}, dC/dt includes:

• Only At e^{-kt}
• Ae^{-kt} – kAt e^{-kt}
• kAt e^{-kt}
• -Ae^{-kt}

Correct Answer: Ae^{-kt} – kAt e^{-kt}

Q44. Differentiation of f(x)=|x| at x=0 yields:

• 0
• Undefined
• 1
• -1

Correct Answer: Undefined

Q45. For optimizing dose, you set derivative of efficacy E'(D)=0; this finds:

• The pharmacokinetic parameter
• Possible optimal dose (extremum)
• The elimination rate constant
• Absolute maximum only

Correct Answer: Possible optimal dose (extremum)

Q46. d/dx [cot x] equals:

• -csc^2 x
• csc^2 x
• -sec^2 x
• sec^2 x

Correct Answer: -csc^2 x

Q47. If f'(x)>0 for all x in interval, f is:

• Decreasing on interval
• Increasing on interval
• Constant on interval
• Concave down

Correct Answer: Increasing on interval

Q48. Use logarithmic differentiation for y=(sin x)^{\tan x}; the method helps because:

• It converts power to product for easier differentiation
• It avoids derivatives entirely
• It converts trig to polynomials
• It integrates the function

Correct Answer: It converts power to product for easier differentiation

Q49. For small h, the difference quotient [f(a+h)-f(a)]/h approximates:

• f(a)/h
• f'(a)
• f”(a)
• Integral from a to a+h

Correct Answer: f'(a)

Q50. In pharmacokinetics, the slope of concentration-time curve at time t equals:

• Total drug administered
• Rate of change of concentration (dC/dt)
• Area under curve (AUC)
• Absolute concentration only

Correct Answer: Rate of change of concentration (dC/dt)