Differentiation – Introduction MCQs With Answer

Differentiation – Introduction MCQs With Answer provides B. Pharm students a focused, practical review of derivatives, rules, and applications in pharmacy. This introduction covers core calculus concepts — power, product, quotient and chain rules — and links them to pharmacokinetics topics like rate of change of drug concentration, elimination constants, Tmax and Cmax. The content is designed for quick revision, concept reinforcement, and exam practice with clear examples and targeted MCQs. Keywords included: Differentiation, derivatives, calculus for pharmacy, pharmacokinetics, B. Pharm MCQs, and answers. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the definition of the derivative of a function f(x) at x?

• The slope of the secant line between two points
• The limit of (f(x+h)-f(x))/h as h approaches 0
• The integral of f(x) over an interval
• The average rate of change over a finite interval

Correct Answer: The limit of (f(x+h)-f(x))/h as h approaches 0

Q2. What is the derivative of f(x) = x^n where n is a constant?

• n * x^(n-1)
• x^(n+1) / (n+1)
• n * x^n
• ln(n) * x^(n-1)

Correct Answer: n * x^(n-1)

Q3. What is the derivative of sin(x)?

• -sin(x)
• cos(x)
• tan(x)
• -cos(x)

Correct Answer: cos(x)

Q4. What is the derivative of e^x with respect to x?

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

Correct Answer: e^x

Q5. Which formula represents the product rule for derivatives of u(x)*v(x)?

• d(uv)/dx = u’v’
• d(uv)/dx = u’v + uv’
• d(uv)/dx = u’ + v’
• d(uv)/dx = uv

Correct Answer: d(uv)/dx = u’v + uv’

Q6. Which formula gives the quotient rule for d(u/v)/dx?

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

Correct Answer: (u’v – uv’)/v^2

Q7. What is the main purpose of the chain rule?

• To differentiate sums of functions
• To integrate composite functions
• To differentiate composite functions f(g(x))
• To calculate limits at infinity

Correct Answer: To differentiate composite functions f(g(x))

Q8. What is the derivative of ln(x)?

• ln(x)/x
• 1/x
• x ln(x)
• e^x

Correct Answer: 1/x

Q9. What does the second derivative f”(x) represent?

• The function average value
• The concavity or rate of change of slope
• The area under f'(x)
• The integral of f(x)

Correct Answer: The concavity or rate of change of slope

Q10. What is the derivative of f(x) = 3x^4?

• 12x^3
• 7x^3
• 3x^5
• 4x^3

Correct Answer: 12x^3

Q11. When is a point called a critical point?

• When f(x) is undefined
• When f'(x) = 0 or f'(x) is undefined
• When f”(x) = 0 only
• When f(x) has a maximum only

Correct Answer: When f'(x) = 0 or f'(x) is undefined

Q12. How is differentiation applied in pharmacokinetics?

• To measure the mass of tablets
• To calculate rate of change of drug concentration over time (dC/dt)
• To determine color of a solution
• To count pills per bottle

Correct Answer: To calculate rate of change of drug concentration over time (dC/dt)

Q13. For first-order elimination dC/dt = -kC, what does k represent?

• Volume of distribution
• Elimination rate constant
• Drug dose
• Maximum concentration

Correct Answer: Elimination rate constant

Q14. Tmax (time to peak concentration) is found by setting which condition?

• C(t) = 0
• dC/dt = 0
• C”(t) = 0
• d^3C/dt^3 = 0

Correct Answer: dC/dt = 0

Q15. What is the derivative of ln(2x)?

• 1/(2x)
• 1/x
• 2/x
• ln2 / x

Correct Answer: 1/x

Q16. What is the derivative of 1/x?

• -1/x^2
• 1/x^2
• -x^2
• 0

Correct Answer: -1/x^2

Q17. Implicit differentiation is used when:

• The function is explicitly y = f(x)
• Variables x and y are related implicitly, e.g., F(x,y)=0
• The function is linear
• The function has no derivative

Correct Answer: Variables x and y are related implicitly, e.g., F(x,y)=0

Q18. Using implicit differentiation on x^2 + y^2 = r^2, dy/dx equals:

• x/y
• -x/y
• y/x
• -y/x

Correct Answer: -x/y

Q19. What is the derivative of tan(x)?

• sec(x)
• sec^2(x)
• csc^2(x)
• tan^2(x)

Correct Answer: sec^2(x)

Q20. Which formula approximates f(x + h) for small h (linear approximation)?

• f(x + h) ≈ f(x) + f'(x) * h
• f(x + h) ≈ f(x) * h
• f(x + h) ≈ f'(x) / h
• f(x + h) ≈ f”(x) * h^2

Correct Answer: f(x + h) ≈ f(x) + f'(x) * h

Q21. The Mean Value Theorem requires which conditions on f(x)?

• Continuous on [a,b] and differentiable on (a,b)
• Differentiable on [a,b] only
• Integrable on [a,b] only
• discontinuous on [a,b]

Correct Answer: Continuous on [a,b] and differentiable on (a,b)

Q22. Rolle’s theorem is a special case of the Mean Value Theorem when:

• f(a) > f(b)
• f(a) = f(b)
• f'(a) = f'(b)
• f is not continuous

Correct Answer: f(a) = f(b)

Q23. If a function is differentiable at a point, what must also be true?

• It is continuous at that point
• It has an integral there
• It is discontinuous
• It has a vertical asymptote

Correct Answer: It is continuous at that point

Q24. Does continuity of f(x) at a point guarantee differentiability there?

• Yes, always
• No, continuity does not guarantee differentiability
• Only for polynomial functions
• Only if f”(x) exists

Correct Answer: No, continuity does not guarantee differentiability

Q25. What is the derivative of e^(2x)?

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

Correct Answer: 2e^(2x)

Q26. The derivative of a^x (a is constant) is:

• a^x
• a^x * ln(a)
• x * a^(x-1)
• ln(x) * a^x

Correct Answer: a^x * ln(a)

Q27. Logarithmic differentiation is particularly useful for:

• Differentiating simple linear functions
• Functions with variable exponents or complicated products
• Integrating rational functions
• Solving algebraic equations

Correct Answer: Functions with variable exponents or complicated products

Q28. What is the derivative of arcsin(x)?

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

Correct Answer: 1/√(1-x^2)

Q29. If C(t) = t^2 * e^{-kt}, what rule(s) do you use to find dC/dt?

• Only power rule
• Only chain rule
• Product rule and chain rule
• Quotient rule only

Correct Answer: Product rule and chain rule

Q30. Is the function f(x) = |x| differentiable at x = 0?

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

Correct Answer: No, not differentiable at x = 0

Q31. What is a partial derivative?

• Derivative of a function of one variable only
• Derivative of a multivariable function with respect to one variable keeping others constant
• The integral of a partial function
• Second derivative of a function

Correct Answer: Derivative of a multivariable function with respect to one variable keeping others constant

Q32. The derivative d(ln C)/dt represents:

• The absolute change in concentration
• The fractional (relative) rate of change of concentration
• The area under concentration-time curve
• The half-life directly

Correct Answer: The fractional (relative) rate of change of concentration

Q33. When performing dimensional analysis, the derivative dC/dt has which units if C is mg/L and t is hours?

• mg·L/hour
• mg/L·hour
• mg·L^-1·hr^-1 (mg/L per hour)
• hr/mg

Correct Answer: mg·L^-1·hr^-1 (mg/L per hour)

Q34. Taylor series expansion around x=a uses derivatives to approximate functions. The linear (first-order) term involves:

• f(a) + f'(a)(x-a)
• f(a) + f”(a)(x-a)^2/2
• Only f”(a)
• Integral of f from a to x

Correct Answer: f(a) + f'(a)(x-a)

Q35. What is the derivative of cos(x)?

• sin(x)
• -sin(x)
• cos(x)
• -cos(x)

Correct Answer: -sin(x)

Q36. Given C(t) = A t / (B + t), what is dC/dt?

• A/(B+t)
• A*B/(B+t)^2
• A*t/(B+t)^2
• 0

Correct Answer: A*B/(B+t)^2

Q37. An inflection point occurs where:

• f'(x) = 0 only
• f”(x) changes sign
• f(x) is undefined
• f”(x) = 0 and f'(x) = 0 always

Correct Answer: f”(x) changes sign

Q38. What is the derivative of √x?

• 1/(2√x)
• √x / 2
• 2√x
• 1/√x

Correct Answer: 1/(2√x)

Q39. Differentiation operator d/dx is linear, meaning d/dx [af + bg] equals:

• a f + b g
• a f’ + b g’
• f’ + g’
• afg’

Correct Answer: a f’ + b g’

Q40. What is d/dx [ln(x^2 + 1)]?

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

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

Q41. How can derivatives help in optimizing drug dose?

• They calculate pill color
• They identify maxima/minima of response curves to find optimal dose
• They measure tablet hardness
• They count the number of excipients

Correct Answer: They identify maxima/minima of response curves to find optimal dose

Q42. In zero-order kinetics, the rate of change of concentration is constant. Which expression matches this?

• dC/dt = -k0 (constant)
• dC/dt = -kC
• dC/dt = kC^2
• dC/dt = 0 only

Correct Answer: dC/dt = -k0 (constant)

Q43. The second derivative is analogous to which mechanical quantity?

• Velocity
• Acceleration
• Displacement
• Force

Correct Answer: Acceleration

Q44. What is the derivative of (sin x)^2?

• 2 sin x cos x
• sin^2 x
• cos^2 x
• 2 cos x

Correct Answer: 2 sin x cos x

Q45. Given f(x) = 3x^3 – 5x + 2, what is f'(2)?

• 31
• 17
• 23
• 11

Correct Answer: 31

Q46. Which finite-difference formula gives a forward approximation of f'(x)?

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

Correct Answer: (f(x+h) – f(x))/h

Q47. Which is true about higher-order derivatives?

• They are derivatives of derivatives, e.g., f”(x) = d/dx(f'(x))
• They are always zero for exponentials
• They cannot exist for polynomials
• They equal integrals

Correct Answer: They are derivatives of derivatives, e.g., f”(x) = d/dx(f'(x))

Q48. Which derivative rule simplifies differentiating y = [g(x)]^n?

• Quotient rule
• Chain rule combined with power rule
• Product rule only
• Mean Value Theorem

Correct Answer: Chain rule combined with power rule

Q49. What is the derivative of ln(C) with respect to time if C follows first-order decay C = C0 e^{-kt}?

• d(ln C)/dt = -k
• d(ln C)/dt = k
• d(ln C)/dt = -C0 k
• d(ln C)/dt = C0 e^{-kt}

Correct Answer: d(ln C)/dt = -k

Q50. When numerically differentiating experimental concentration data, a smaller h in finite differences generally:

• Always increases error
• Reduces truncation error but may increase rounding error
• Has no effect on error
• Eliminates measurement noise

Correct Answer: Reduces truncation error but may increase rounding error

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