Derivative of a function MCQs With Answer

Understanding the Derivative of a function MCQs With Answer is essential for B. Pharm students studying calculus applied to pharmacy. This focused introduction highlights core concepts—definition via limits, differentiation rules (power, product, quotient, chain), implicit differentiation, higher-order derivatives, and derivatives of polynomial, trigonometric, exponential and logarithmic functions. Emphasis is placed on pharmacokinetic applications such as rate of change of drug concentration, slope of concentration–time curves, maxima/minima for peak concentrations, and inflection points for changing dynamics. These targeted MCQs with answers build problem-solving skills for dosage calculations, modeling and exam preparation. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the definition of the derivative of f at x?

• The limit as h approaches 0 of [f(x+h)-f(x)]/h
• The integral of f from 0 to x
• The average value of f over an interval
• The second derivative of f at x

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

Q2. What is the derivative of f(x) = x^3?

• 3x^2
• x^2
• 3x^3
• x

Correct Answer: 3x^2

Q3. What is d/dx [sin x]?

• cos x
• -cos x
• sin x
• -sin x

Correct Answer: cos x

Q4. What is d/dx [e^{2x}]?

• 2 e^{2x}
• e^{2x}
• e^{x}
• 2 e^{x}

Correct Answer: 2 e^{2x}

Q5. What is d/dx [ln x] for x>0?

• 1/x
• ln x
• x
• 1/ln x

Correct Answer: 1/x

Q6. Which formula is the product rule for u(x) and v(x)?

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

Correct Answer: u’v + uv’

Q7. What is the quotient rule for d/dx [u/v]?

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

Correct Answer: (u’v – uv’) / v^2

Q8. Which expression correctly states the chain rule?

• d/dx f(g(x)) = f'(g(x)) g'(x)
• d/dx f(g(x)) = f(g'(x))
• d/dx f(g(x)) = f'(x) g'(x)
• d/dx f(g(x)) = g'(f(x))

Correct Answer: d/dx f(g(x)) = f'(g(x)) g'(x)

Q9. What is d/dx [(3x+2)^4]?

• 12(3x+2)^3
• (3x+2)^3
• 4(3x+2)^3
• 36(3x+2)^3

Correct Answer: 12(3x+2)^3

Q10. What is d/dx [tan x]?

• sec^2 x
• csc^2 x
• tan x
• 1/cos x

Correct Answer: sec^2 x

Q11. Which notation represents the second derivative of y with respect to x?

• d^2y/dx^2
• dy/dx^2
• d(y^2)/dx
• d^2x/dy^2

Correct Answer: d^2y/dx^2

Q12. What is d/dx [1/x] for x≠0?

• -1/x^2
• 1/x^2
• 0
• 1

Correct Answer: -1/x^2

Q13. What is d/dx [sqrt(x)] for x>0?

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

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

Q14. What is d/dx [arctan x]?

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

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

Q15. What is the derivative of a constant function c?

• 0
• c
• 1
• Undefined

Correct Answer: 0

Q16. If f(x) = e^{ax} with constant a, what is f'(x)?

• a e^{ax}
• e^{ax}
• a x e^{ax}
• ln(a) e^{ax}

Correct Answer: a e^{ax}

Q17. Given xy = 1, what is dy/dx by implicit differentiation?

• -y/x
• y/x
• -x/y
• 1/(xy)

Correct Answer: -y/x

Q18. What is d/dx [ln(3x)] for x>0?

• 1/x
• 1/(3x)
• 3/x
• ln 3 / x

Correct Answer: 1/x

Q19. What is d/dx [sin^2 x]?

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

Correct Answer: 2 sin x cos x

Q20. What is d/dx [cos x]?

• -sin x
• sin x
• -cos x
• cos x

Correct Answer: -sin x

Q21. In pharmacokinetics, what does the derivative dC/dt represent?

• Rate of change of drug concentration with respect to time
• Total drug amount in the body
• Half-life of the drug

Correct Answer: Rate of change of drug concentration with respect to time

Q22. What is the derivative of |x| for x>0?

• 1
• -1
• 0
• Undefined

Correct Answer: 1

Q23. What is the slope of the tangent to f(x)=x^2 at x=2?

• 4
• 2
• 8
• 1

Correct Answer: 4

Q24. What is d/dx [ln(x^2 + 1)]?

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

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

Q25. What is d/dx [sec x]?

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

Correct Answer: sec x tan x

Q26. Which rule applies to d/dx [c f(x)] where c is constant?

• c f'(x)
• f'(x)/c
• c’ f(x)
• f(x)

Correct Answer: c f'(x)

Q27. If y = f^{-1}(x) is the inverse of f, what is dy/dx?

• 1 / f'(f^{-1}(x))
• f'(x)
• 1 / f(x)
• f^{-1}'(x)

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

Q28. What is d/dx [x^n] for any real n?

• n x^{n-1}
• x^{n+1}/n
• x^n
• n^x

Correct Answer: n x^{n-1}

Q29. Which statement is true?

• If f is differentiable at a point then f is continuous at that point
• If f is continuous then f is differentiable
• Continuity and differentiability are unrelated
• A function can be differentiable but not continuous

Correct Answer: If f is differentiable at a point then f is continuous at that point

Q30. What is d/dx [5^x]?

• ln 5 * 5^x
• 5^x
• x 5^{x-1}
• 5 ln x

Correct Answer: ln 5 * 5^x

Q31. What is d/dx [arccos x]?

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

Correct Answer: -1 / sqrt(1 – x^2)

Q32. What is d/dx [x^2 sin x]?

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

Correct Answer: 2x sin x + x^2 cos x

Q33. What is d/dx [ln(sin x)] for 0

• cot x
• tan x
• csc x
• sec x

Correct Answer: cot x

Q34. If C(t) = C0 e^{-kt}, what is dC/dt?

• -k C0 e^{-kt}
• k C0 e^{-kt}
• -C0 e^{-kt}
• 0

Correct Answer: -k C0 e^{-kt}

Q35. If f'(x0)=0 and f”(x0)<0, what can be concluded at x0?

• f has a local maximum at x0
• f has a local minimum at x0
• f is inflection at x0
• Nothing can be concluded

Correct Answer: f has a local maximum at x0

Q36. Simplify and differentiate: f(x) = (3x^2 + 2x)/x.

• f'(x) = 3
• f'(x) = 6x + 2
• f'(x) = 3x + 2
• f'(x) = 1/x

Correct Answer: f'(x) = 3

Q37. What is d/dx [1/(1+x^2)]?

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

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

Q38. What does the Leibniz notation d/dx represent?

• Derivative with respect to x
• Integral with respect to x
• Product of functions
• Limit of a sequence

Correct Answer: Derivative with respect to x

Q39. Which conditions are required for the Mean Value Theorem?

• Continuity on [a,b] and differentiability on (a,b)
• Differentiability on [a,b] only
• Continuity on (a,b) only
• Function must be linear

Correct Answer: Continuity on [a,b] and differentiability on (a,b)

Q40. What is d/dx [x sin x] evaluated at x=0?

• 0
• 1
• sin 0
• cos 0

Correct Answer: 0

Q41. For f(x)=x^3 – 6x^2 + 9x, where are the critical points (f’=0)?

• x = 1 and x = 3
• x = 0 and x = 3
• x = -1 and x = 3
• x = 1 only

Correct Answer: x = 1 and x = 3

Q42. What does the sign of the second derivative indicate?

• Concavity of the function
• Value of the function
• Domain of the function
• Periodicity of the function

Correct Answer: Concavity of the function

Q43. What is d/dx [e^{x^2}]?

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

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

Q44. If A(t) = k t^n where k and n are constants, what is dA/dt?

• n k t^{n-1}
• k t^n
• k n t^{n}
• k t^{n-1}

Correct Answer: n k t^{n-1}

Q45. How is the peak concentration time determined from C(t)?

• Set dC/dt = 0 and solve for t
• Set C(t) = 0 and solve for t
• Find where C(t) is minimum
• Differentiate C(t) twice

Correct Answer: Set dC/dt = 0 and solve for t

Q46. If f'(x)=0 for all x in an interval, what is f(x) on that interval?

• Constant
• Linear
• Quadratic
• Undefined

Correct Answer: Constant

Q47. What is d/dx [1]?

• 0
• 1
• -1
• Undefined

Correct Answer: 0

Q48. What is d/dx [log_a x] where a>0, a≠1?

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

Correct Answer: 1/(x ln a)

Q49. If f(x)=x^2 sin x and you apply the product rule, which term appears?

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

Correct Answer: x^2 cos x

Q50. What is f'(3) for f(x)=x^2 using the limit definition or power rule?

• 6
• 3
• 9
• 12

Correct Answer: 6

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