Derivative of a function MCQs With Answer

Introduction: The derivative of a function is a core calculus concept for B. Pharm students, linking mathematics to pharmacokinetics and drug formulation. Understanding differentiation helps analyze rates of change in concentration-time profiles, absorption and elimination rates, and slope-based optimization in drug delivery. This SEO-friendly guide covers differentiation rules — power, product, quotient, chain, logarithmic and implicit differentiation — with applied examples relevant to pharmacy. Practice with targeted MCQs improves problem-solving for kinetics, dose-response curves, and mathematical modeling in pharmaceutics. Keywords: derivative of a function, MCQs with answer, B. Pharm students, calculus for pharmacy, rate of change, differentiation rules, pharmacokinetics. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the derivative of a function at a point?

• The slope of the tangent to the curve at that point
• The area under the curve at that point
• The average slope between two points
• The integral from 0 to that point

Correct Answer: The slope of the tangent to the curve at that point

Q2. Which formula gives the derivative of x^n for constant n?

• d/dx (x^n) = n x^(n-1)
• d/dx (x^n) = x^n / n
• d/dx (x^n) = n x^(n+1)
• d/dx (x^n) = (x^(n-1))/n

Correct Answer: d/dx (x^n) = n x^(n-1)

Q3. The derivative of sin x with respect to x is:

• cos x
• -cos x
• tan x
• sec x

Correct Answer: cos x

Q4. What is d/dx of e^(kx), where k is a constant?

• k e^(kx)
• e^(kx) / k
• k x e^(kx)
• e^(k) x

Correct Answer: k e^(kx)

Q5. The product rule for derivatives of u(x)v(x) is:

• (u v)’ = u’ v + u v’
• (u v)’ = u’ v’
• (u v)’ = u v’ – u’ v
• (u v)’ = u’ + v’

Correct Answer: (u v)’ = u’ v + u v’

Q6. The quotient rule for (u/v)’ equals:

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

Correct Answer: (u’ v – u v’) / v^2

Q7. The chain rule for d/dx [f(g(x))] is:

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

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

Q8. The derivative of ln x is:

• 1/x
• ln x
• x ln x
• x

Correct Answer: 1/x

Q9. What is a higher-order derivative?

• A derivative of a derivative (e.g., second derivative)
• A derivative evaluated at a larger x
• A derivative multiplied by a constant
• An integral of the derivative

Correct Answer: A derivative of a derivative (e.g., second derivative)

Q10. For first-order elimination dC/dt = -k C, the elimination rate constant k equals:

• – (1/C) dC/dt
• (1/C) dC/dt
• -C dC/dt
• dC/dt

Correct Answer: – (1/C) dC/dt

Q11. Critical points for local maxima or minima occur when:

• The first derivative is zero or undefined
• The function value is zero
• The second derivative is negative
• The integral is maximum

Correct Answer: The first derivative is zero or undefined

Q12. An inflection point is identified when:

• The second derivative changes sign
• The first derivative equals zero
• The function equals zero
• The derivative is infinite

Correct Answer: The second derivative changes sign

Q13. Units of a derivative dC/dt when C is mg/L and t is h are:

• mg·L^-1·h^-1
• mg·L·h
• mg/L
• h/mg

Correct Answer: mg·L^-1·h^-1

Q14. d/dx [ln(f(x))] equals:

• f'(x)/f(x)
• ln f(x) f'(x)
• f(x)/f'(x)
• f'(x) ln f(x)

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

Q15. The derivative of tan x is:

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

Correct Answer: sec^2 x

Q16. The derivative of arcsin x is:

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

Correct Answer: 1 / sqrt(1 – x^2)

Q17. d/dx (1/x) is:

• -1/x^2
• 1/x^2
• -1/x
• 0

Correct Answer: -1/x^2

Q18. The derivative of x^(1/2) is:

• 1 / (2 sqrt(x))
• 1 / (sqrt(x))
• sqrt(x)/2
• 2 sqrt(x)

Correct Answer: 1 / (2 sqrt(x))

Q19. In pharmacokinetics, the slope of a concentration-time curve at time t indicates:

• Instantaneous rate of change in drug concentration
• Total amount of drug absorbed
• Area under the curve (AUC)
• Bioavailability percentage

Correct Answer: Instantaneous rate of change in drug concentration

Q20. The tangent line approximates a function near a point because:

• The derivative gives the best linear approximation locally
• The integral equals the tangent slope
• The function is constant near the point
• The second derivative is zero

Correct Answer: The derivative gives the best linear approximation locally

Q21. The forward difference approximation for a derivative is:

• f'(x) ≈ (f(x+h) – f(x)) / h
• f'(x) ≈ (f(x) – f(x-h)) / h
• f'(x) ≈ (f(x+h) – f(x-h)) / (2h)
• f'(x) ≈ f(x) / h

Correct Answer: f'(x) ≈ (f(x+h) – f(x)) / h

Q22. d/dx [log_a x] equals:

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

Correct Answer: 1 / (x ln a)

Q23. d/dx [e^(x^2)] is:

• 2x e^(x^2)
• e^(x^2)
• x e^(x^2)
• 2 e^(x^2)

Correct Answer: 2x e^(x^2)

Q24. d/dx [x ln x] equals:

• 1 + ln x
• ln x
• x / ln x
• 1 / x

Correct Answer: 1 + ln x

Q25. The derivative of arctan x is:

• 1 / (1 + x^2)
• 1 / sqrt(1 – x^2)
• -1 / (1 + x^2)
• ln(1 + x^2)

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

Q26. If C(t) = C0 e^(-k t), then dC/dt equals:

• -k C0 e^(-k t)
• k C0 e^(-k t)
• -C0 e^(k t)
• C0 e^(-k t)/k

Correct Answer: -k C0 e^(-k t)

Q27. Implicit differentiation of x^2 + y^2 = r^2 gives dy/dx =:

• -x / y
• x / y
• y / x
• -y / x

Correct Answer: -x / y

Q28. The function f(x)=|x| is differentiable at x=0?

• No, it is not differentiable at x=0
• Yes, derivative is 0 at x=0
• Yes, derivative is 1 at x=0
• Yes, derivative is -1 at x=0

Correct Answer: No, it is not differentiable at x=0

Q29. The Mean Value Theorem guarantees existence of c in (a,b) such that:

• f'(c) = [f(b) – f(a)] / (b – a)
• f(c) = [f(b) + f(a)] / 2
• f”(c) = 0
• f'(c) = 0

Correct Answer: f'(c) = [f(b) – f(a)] / (b – a)

Q30. The second derivative f”(x) provides information about:

• Concavity of the function
• The function’s integral
• Only the slope sign
• Whether the function is periodic

Correct Answer: Concavity of the function

Q31. The derivative of sinh x is:

• cosh x
• sinh x
• tanh x
• sech x

Correct Answer: cosh x

Q32. d/dt [ln C(t)] equals:

• (1/C) dC/dt
• ln C * dC/dt
• C / dC/dt
• dC/dt

Correct Answer: (1/C) dC/dt

Q33. The time of maximum concentration (Tmax) for oral dosing is found by:

• Setting dC/dt = 0 and solving for t
• Maximizing C directly without derivatives
• Setting C = 0
• Integrating C over time

Correct Answer: Setting dC/dt = 0 and solving for t

Q34. Derivative of product u v w equals:

• u’ v w + u v’ w + u v w’
• u’ v’ w’
• (u v w)’ = u v w
• u’ v w – u v’ w + u v w’

Correct Answer: u’ v w + u v’ w + u v w’

Q35. d/dx [sin x / x] simplifies to:

• (x cos x – sin x) / x^2
• (sin x – x cos x) / x^2
• (cos x) / x
• (x sin x – cos x) / x^2

Correct Answer: (x cos x – sin x) / x^2

Q36. d/dx [x^x] equals:

• x^x (ln x + 1)
• x^(x-1)
• x^x ln x
• e^(x ln x)

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

Q37. L’Hôpital’s rule is used to evaluate limits of indeterminate forms like:

• 0/0 or ∞/∞ by differentiating numerator and denominator
• Any infinite product
• Only polynomial limits
• Discrete sequences only

Correct Answer: 0/0 or ∞/∞ by differentiating numerator and denominator

Q38. d/dx (3x^4) equals:

• 12x^3
• 3x^3
• 7x^3
• 12x^4

Correct Answer: 12x^3

Q39. If f(x)=x^2, f'(2) equals:

• 4
• 2
• 8
• 1

Correct Answer: 4

Q40. d/dx [1/(1+x^2)] equals:

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

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

Q41. The derivative of arccos x is:

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

Correct Answer: -1 / sqrt(1 – x^2)

Q42. By the Fundamental Theorem of Calculus, d/dx ∫_0^x f(t) dt equals:

• f(x)
• ∫_0^x f'(t) dt
• 0
• f(0)

Correct Answer: f(x)

Q43. d/dx [(ln x)^2] equals:

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

Correct Answer: 2 ln x / x

Q44. d/dx [e^x sin x] equals:

• e^x sin x + e^x cos x
• e^x (sin x – cos x)
• e^x sin x
• e^x cos x

Correct Answer: e^x sin x + e^x cos x

Q45. d/dx [sqrt(1 + x^2)] equals:

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

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

Q46. The derivative of a constant is:

• 0
• The constant itself
• 1
• Undefined

Correct Answer: 0

Q47. If f'(c)=0 and f”(c)>0, then c is:

• A local minimum
• A local maximum
• An inflection point
• Not a critical point

Correct Answer: A local minimum

Q48. d/dx [e^{sin x}] equals:

• cos x · e^{sin x}
• sin x · e^{sin x}
• e^{sin x}
• cos x + e^{sin x}

Correct Answer: cos x · e^{sin x}

Q49. To estimate the initial rate of drug disappearance at t=0, you compute:

• dC/dt at t=0
• Integral of C from 0 to infinity
• Average concentration over time
• C(0) divided by total time

Correct Answer: dC/dt at t=0

Q50. If C(t) = A(t) · B(t), then dC/dt equals:

• A dB/dt + B dA/dt
• A dB/dt – B dA/dt
• A’ B’
• A B

Correct Answer: A dB/dt + B dA/dt