Understanding the derivative of a function is essential for B. Pharm students studying pharmacokinetics, drug absorption and elimination. This SEO-friendly guide on “Derivative of a function MCQs With Answer” explains differentiation concepts—limits, rules (product, quotient, chain), higher-order derivatives, and applications to concentration-time profiles—in clear, pharmacy-centric language. Familiarity with derivatives helps interpret rate of change, Cmax, elimination rate constants and dose-response curves. These targeted MCQs with answers reinforce conceptual understanding, problem-solving skills and real-world pharmacy examples. Practice improves speed and accuracy for exams and clinical calculations. Key keyword coverage includes differentiation rules, derivative formulas, first principles, implicit differentiation and applications in pharmacy math. Now let’s test your knowledge with 50 MCQs on this topic.
Q1. What is the definition of the derivative f'(x) of a function f at x?
- The value of f(x) at x
- The limit of [f(x+h) – f(x)]/h as h approaches 0
- The integral of f from 0 to x
- The average value of f over an interval
Correct Answer: The limit of [f(x+h) – f(x)]/h as h approaches 0
Q2. Which rule is used to differentiate the product of two functions u(x) and v(x)?
- Quotient rule
- Chain rule
- Product rule
- Power rule
Correct Answer: Product rule
Q3. What is the derivative of f(x) = x^n for any real n (power rule)?
- n x^(n-1)
- x^(n+1)/(n+1)
- n x^n
- x^(n-1)
Correct Answer: n x^(n-1)
Q4. Using first principles, what is the derivative of f(x) = x^2?
- 2x
- x
- 2
- x^2
Correct Answer: 2x
Q5. Which formula correctly expresses the quotient rule for f(x)=u/v?
- (u’v – uv’)/v^2
- (u’v + uv’)/v
- (u’v + uv’)/v^2
- (u’v – uv’)/v
Correct Answer: (u’v – uv’)/v^2
Q6. What is the derivative of e^{ax}, where a is a constant?
- a e^{ax}
- e^{ax}/a
- e^x
- ae^x
Correct Answer: a e^{ax}
Q7. Which rule do you apply to differentiate f(x)=sin(3x^2)?
- Quotient rule
- Product rule
- Chain rule
- Integration by parts
Correct Answer: Chain rule
Q8. What is the derivative of ln(x)?
- 1/x
- ln(x)/x
- x ln(x)
- e^x
Correct Answer: 1/x
Q9. For a drug concentration C(t)=C0 e^{-kt}, what is dC/dt?
- k C0 e^{-kt}
- -k C0 e^{-kt}
- -C0 e^{kt}
- k e^{-kt}
Correct Answer: -k C0 e^{-kt}
Q10. If f(x)=sin x, what is f'(x)?
- cos x
- -sin x
- sin x
- -cos x
Correct Answer: cos x
Q11. Which statement is true: differentiability implies continuity, or continuity implies differentiability?
- Continuity implies differentiability
- Differentiability implies continuity
- Neither implies the other
- Both imply each other
Correct Answer: Differentiability implies continuity
Q12. What is the derivative of f(x)=tan x?
- sec^2 x
- cos^2 x
- 1/cos x
- csc^2 x
Correct Answer: sec^2 x
Q13. Using the chain rule, what is d/dx [ln(sin x)]?
- cos x / sin x
- sin x / cos x
- 1 / (sin x)
- cos x
Correct Answer: cos x / sin x
Q14. For f(x)=a^x where a>0, what is f'(x)?
- a^x ln a
- a^{x+1}
- x a^{x-1}
- ln x
Correct Answer: a^x ln a
Q15. What is the derivative of 1/x?
- -1/x^2
- 1/x^2
- -x
- 1
Correct Answer: -1/x^2
Q16. If C(t) is concentration and dC/dt = 0 at t = t*, what does this indicate about the concentration at t*?
- It is a local extremum (possible Cmax or Cmin)
- The function is discontinuous at t*
- The concentration is zero
- The elimination rate is infinite
Correct Answer: It is a local extremum (possible Cmax or Cmin)
Q17. Which derivative gives the rate of change of the slope (concavity) of a function?
- First derivative
- Second derivative
- Third derivative
- Zero derivative
Correct Answer: Second derivative
Q18. What is the second derivative of f(x)=sin x?
- -sin x
- cos x
- -cos x
- sin x
Correct Answer: -sin x
Q19. Which derivative rule helps differentiate f(x) = (3x^2 + 2x)(x – 1)?
- Chain rule
- Product rule
- Quotient rule
- Integration rule
Correct Answer: Product rule
Q20. What is d/dx [arctan x]?
- 1/(1+x^2)
- 1/(1-x^2)
- 1/(1+x)
- x/(1+x^2)
Correct Answer: 1/(1+x^2)
Q21. If drug plasma concentration C(t) follows C0 e^{-kt}, how is half-life t1/2 related to k?
- t1/2 = k/ln 2
- t1/2 = ln 2 / k
- t1/2 = 2k
- t1/2 = 1/(k ln 2)
Correct Answer: t1/2 = ln 2 / k
Q22. Which of the following is the derivative of f(x)=cos x?
- sin x
- -sin x
- -cos x
- cos x
Correct Answer: -sin x
Q23. What is the derivative of f(x)=ln(ax) where a is constant and x>0?
- 1/(ax)
- 1/x
- a/x
- ln a / x
Correct Answer: 1/x
Q24. Which statement best describes an inflection point?
- Point where function is discontinuous
- Point where first derivative is zero and function has maximum
- Point where concavity changes sign (second derivative changes sign)
- Point where function equals zero
Correct Answer: Point where concavity changes sign (second derivative changes sign)
Q25. For f(x)=x^(1/2), what is f'(x)?
- 1/(2 sqrt(x))
- sqrt(x)/2
- 1/(sqrt(x))
- 2 sqrt(x)
Correct Answer: 1/(2 sqrt(x))
Q26. When differentiating implicitly given F(x,y)=0, what derivative do you compute to find dy/dx?
- Differentiate both sides with respect to y only
- Differentiate both sides with respect to x treating y as implicit function of x
- Integrate both sides
- Use product rule only
Correct Answer: Differentiate both sides with respect to x treating y as implicit function of x
Q27. If f(x)=3x^4 – 5x + 2, what is f'(x)?
- 12x^3 – 5
- 12x^3 – 5x
- 3x^3 – 5
- 12x^4 – 5
Correct Answer: 12x^3 – 5
Q28. What is the derivative of f(x)=ln(e^{2x}) simplified?
- 2
- e^{2x}
- ln 2
- 2 e^{2x}
Correct Answer: 2
Q29. Which method approximates the derivative using f(x+h)-f(x) over h for small h?
- Backward differentiation
- Forward difference (finite difference)
- Simpson’s rule
- Trapezoidal rule
Correct Answer: Forward difference (finite difference)
Q30. For pharmacokinetic curve C(t), the time of peak concentration Cmax occurs when:
- dC/dt > 0
- dC/dt < 0
- dC/dt = 0 and changes from positive to negative
- C(t) = 0
Correct Answer: dC/dt = 0 and changes from positive to negative
Q31. What is d/dx [x^(-3)]?
- -3 x^{-4}
- 3 x^{-2}
- -x^{-2}
- -3 x^{-2}
Correct Answer: -3 x^{-4}
Q32. Which derivative rule is most useful for f(x) = (3x+1)^5?
- Quotient rule
- Power rule combined with chain rule
- Product rule
- Logarithmic differentiation
Correct Answer: Power rule combined with chain rule
Q33. What is d/dx [sin^{-1}(x)] for -1
- 1/sqrt(1-x^2)
- 1/(1-x^2)
- sqrt(1-x^2)
- -1/sqrt(1-x^2)
Correct Answer: 1/sqrt(1-x^2)
Q34. Which property expresses linearity of differentiation?
- d/dx [f + g] = f’ g’
- d/dx [c f(x)] = c f'(x) and d/dx [f+g] = f'(x)+g'(x)
- d/dx [f g] = f’ + g’
- d/dx [f/g] = f’/g’
Correct Answer: d/dx [c f(x)] = c f'(x) and d/dx [f+g] = f'(x)+g'(x)
Q35. If concentration C(t)=At/(B+t), what mathematical technique is commonly used to find dC/dt?
- Chain rule only
- Quotient rule
- Product rule only
- Integration
Correct Answer: Quotient rule
Q36. What is d/dx [e^{x^2}]?
- 2x e^{x^2}
- x e^{x^2}
- e^{x^2}
- 2 e^{x^2}
Correct Answer: 2x e^{x^2}
Q37. For f(x)=x|x|, is f differentiable at x=0?
- Yes, derivative is 0
- No, because left and right derivatives differ
- Yes, derivative is 1
- Undefined because function not continuous
Correct Answer: No, because left and right derivatives differ
Q38. When applying logarithmic differentiation, which functions benefit most from this technique?
- Simple polynomials like x^2
- Products and powers where variable appears in both base and exponent
- Constant functions
- Pure exponentials with constant base and exponent
Correct Answer: Products and powers where variable appears in both base and exponent
Q39. What is the derivative of f(x)=sec x?
- sec x tan x
- sec^2 x
- tan x
- csc x tan x
Correct Answer: sec x tan x
Q40. If the first derivative is positive on an interval, the function is:
- Decreasing on that interval
- Increasing on that interval
- Concave down on that interval
- Has an inflection point on that interval
Correct Answer: Increasing on that interval
Q41. How do you compute dy/dx for implicit relation x^2 + y^2 = 25?
- Differentiate both sides: 2x + 2y dy/dx = 0, so dy/dx = -x/y
- dy/dx = x/y
- dy/dx = -2x/(2y) = -x/y^2
- dy/dx = 0
Correct Answer: Differentiate both sides: 2x + 2y dy/dx = 0, so dy/dx = -x/y
Q42. Which derivative corresponds to the acceleration if position s(t) is given?
- First derivative s'(t)
- Second derivative s”(t)
- Third derivative s”'(t)
- Integral of s(t)
Correct Answer: Second derivative s”(t)
Q43. For f(x)=ln(x^2+1), what is f'(x)?
- 2x/(x^2+1)
- 1/(x^2+1)
- ln(2x)/(x^2+1)
- x/(x^2+1)
Correct Answer: 2x/(x^2+1)
Q44. What is d/dx [x sin x]?
- sin x + x cos x
- x cos x
- cos x + x sin x
- sin x
Correct Answer: sin x + x cos x
Q45. Which theorem guarantees that there exists a c in (a,b) such that f'(c) = [f(b)-f(a)]/(b-a) for continuous f on [a,b] and differentiable on (a,b)?
- Rolle’s Theorem
- Mean Value Theorem
- Intermediate Value Theorem
- Fundamental Theorem of Calculus
Correct Answer: Mean Value Theorem
Q46. What is the derivative of f(x)=log_{10}(x)?
- 1/(x ln 10)
- 1/x
- ln 10 / x
- 10^x
Correct Answer: 1/(x ln 10)
Q47. If velocity v(t)=ds/dt, and v'(t)>0, what does this indicate about motion?
- Speed is decreasing
- Acceleration is positive (speed increasing)
- Object is stationary
- Displacement is negative
Correct Answer: Acceleration is positive (speed increasing)
Q48. Which expression is the derivative of f(x)= (x^2 + 1)^{3} using chain rule?
- 3(x^2+1)^2 * 2x
- 6(x^2+1)^3
- 3(x^2+1)^2
- 2x (x^2+1)^3
Correct Answer: 3(x^2+1)^2 * 2x
Q49. What are the units of derivative dC/dt if C is mg/L and t is hours?
- mg·L/hour
- mg/L·hour
- mg·L/hours^2
- mg·L per hour (mg·L-1·hr-1)
Correct Answer: mg·L per hour (mg·L-1·hr-1)
Q50. Newton’s method for finding roots requires which of the following at each iteration?
- Second derivative value only
- Function value and its first derivative at current iterate
- Integral of function
- Only the function value
Correct Answer: Function value and its first derivative at current iterate
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