Derivative of a function MCQs With Answer

Derivative of a function MCQs With Answer are essential for B.Pharm students who need to master calculus concepts applied to pharmacokinetics, drug release, and rate processes. This concise, SEO-friendly introduction covers core ideas like the derivative as the instantaneous rate of change, rules (power, product, quotient, chain), higher-order derivatives, and real-life applications such as concentration-time profiles, half-life, and dose–response curves. Clear understanding of derivatives helps in modeling drug absorption, elimination, and reaction kinetics, improving problem-solving in pharmaceutics and biopharmaceutics. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What does the derivative of a function at a point represent?

• The average value of the function near the point
• The instantaneous rate of change of the function at that point
• The area under the curve up to that point
• The maximum value of the function

Correct Answer: The instantaneous rate of change of the function at that point

Q2. If C(t) = C0 e^-kt describes drug concentration, what is dC/dt?

• dC/dt = k C0 e^-kt
• dC/dt = -k C0 e^-kt
• dC/dt = -C0 e^-kt
• dC/dt = k e^-kt

Correct Answer: dC/dt = -k C0 e^-kt

Q3. What is the derivative of f(x)=xⁿ (power rule) for n constant?

• f'(x)=n x^n-1
• f'(x)=xⁿ⁺¹/n
• f'(x)=n xⁿ⁺¹
• f'(x)=xⁿ ln n

Correct Answer: f'(x)=n x^n-1

Q4. For f(x)=ln(x), what is f'(x)?

• 1/ln(x)
• 1/x
• ln(x)/x
• x

Correct Answer: 1/x

Q5. Which rule is used to differentiate f(x)=u(x)v(x)?

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

Correct Answer: Product rule

Q6. If f(x)=u(x)/v(x), which formula gives f'(x)?

• (u’v – uv’)/v
• (u’v + uv’)/v²
• (u’v – uv’)/v²
• (u’v + uv’)/v

Correct Answer: (u’v – uv’)/v²

Q7. Find derivative of f(x)=e^ax where a is constant.

• a e^ax
• e^ax/a
• ln(a) e^ax
• e^x

Correct Answer: a e^ax

Q8. For y = sin(x), y’ equals?

• cos(x)
• -cos(x)
• sin(x)
• -sin(x)

Correct Answer: cos(x)

Q9. Using chain rule, derivative of y = (3x+1)⁴ is:

• 4(3x+1)³
• 12(3x+1)³
• (3x+1)⁴·3
• 4(3x+1)⁴

Correct Answer: 12(3x+1)³

Q10. If concentration follows C(t)=At/(Bt+1), what derivative rule is most directly applied?

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

Correct Answer: Quotient rule

Q11. The derivative of pH = -log₁₀[H+] with respect to [H+] is:

• -1/[H+]
• -1/(ln10 · [H+])
• 1/(ln10 · [H+])
• 1/[H+]

Correct Answer: -1/(ln10 · [H+])

Q12. If V(t)=Vmax·S/(Km+S) (Michaelis-Menten), dV/dS involves:

• Vmax·(Km+S)/(Km+S)²
• Vmax·Km/(Km+S)²
• Vmax·Km/(Km+S)
• Vmax·(Km)/(S)

Correct Answer: Vmax·Km/(Km+S)²

Q13. What is the derivative of f(x)=arctan(x)?

• 1/(1+x²)
• 1/(1-x²)
• x/(1+x²)
• 1/√(1-x²)

Correct Answer: 1/(1+x²)

Q14. The second derivative f”(x) gives information on:

• Instantaneous rate of change
• Concavity and acceleration
• Average slope over interval
• Value of function

Correct Answer: Concavity and acceleration

Q15. If f'(x0)=0 and f”(x0)<0, x0 is:

• A point of inflection
• Local minimum
• Local maximum
• Not differentiable

Correct Answer: Local maximum

Q16. The derivative of |x| at x=0 is:

• 0
• 1
• -1
• Does not exist

Correct Answer: Does not exist

Q17. For parametric curves x(t), y(t), dy/dx equals:

• dx/dt ÷ dy/dt
• dy/dt · dx/dt
• (dy/dt)/(dx/dt)
• Integral of dy/dt

Correct Answer: (dy/dt)/(dx/dt)

Q18. Differentiate f(x)=ln(ax) where a>0 constant.

• 1/(ax)
• a/(ax)
• 1/x
• ln(a)/x

Correct Answer: 1/x

Q19. If drug amount A(t)=kt, what is dA/dt?

• A(t)/t
• k
• t/k
• 0

Correct Answer: k

Q20. The derivative of tan(x) is:

• sec(x)
• sec²(x)
• cos²(x)
• csc²(x)

Correct Answer: sec²(x)

Q21. Use implicit differentiation: For x² + y² = R², dy/dx =

• -x/y
• -y/x
• x/y
• y/x

Correct Answer: -x/y

Q22. The derivative of f(x)=x·e^x is:

• e^x + x e^x
• x e^x
• e^x
• xe^2x

Correct Answer: e^x + x e^x

Q23. If C(t)=C0/(1+kt) (simple elimination), dC/dt at t=0 equals:

• -C0 k
• -C0 k/(1+kt)²
• -C0 k at t=0 gives -C0 k
• 0

Correct Answer: -C0 k

Q24. The derivative of f(x)=log₁₀(x) is:

• 1/(x ln 10)
• 1/x
• ln 10 / x
• ln(x)/10

Correct Answer: 1/(x ln 10)

Q25. Which expression equals derivative of inverse function (f^-1)'(y)?

• 1/f'(y)
• 1/f'(f^-1(y))
• f'(f^-1(y))
• -1/f'(f^-1(y))

Correct Answer: 1/f'(f^-1(y))

Q26. For f(x)=x³-3x, critical points where f’=0 are:

• x=0, x=±1
• x=1 only
• x=0 only
• No real critical points

Correct Answer: x=0, x=±1

Q27. Differentiate f(x)=sin²(x) using chain rule.

• 2 sin(x) cos(x)
• sin(2x)
• cos(2x)
• Both 2 sin(x) cos(x) and sin(2x) are equivalent

Correct Answer: Both 2 sin(x) cos(x) and sin(2x) are equivalent

Q28. If concentration C(t)=C0 e^{-k t} and k increases, instantaneous elimination rate dC/dt at fixed t:

• Decreases in magnitude
• Increases in magnitude (more negative)
• Remains unchanged
• Becomes positive

Correct Answer: Increases in magnitude (more negative)

Q29. The derivative of f(x)=x^1/2 is:

• 1/(2√x)
• √x/2
• 1/√x
• x^-1/2/2

Correct Answer: 1/(2√x)

Q30. Using logarithmic differentiation, derivative of f(x)=x^x is:

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

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

Q31. What is the derivative of f(x)=sec(x)?

• sec(x)tan(x)
• sec(x)cos(x)
• tan(x)
• sec²(x)

Correct Answer: sec(x)tan(x)

Q32. If drug absorption A(t)=t e^-t, what is dA/dt?

• e^-t(1 – t)
• e^-t(1 + t)
• t e^-t
• -t e^-t

Correct Answer: e^-t(1 – t)

Q33. For composite f(g(x)), the chain rule form is:

• f'(g(x)) · g'(x)
• g'(f(x)) · f'(x)
• f(g'(x))
• f'(x) + g'(x)

Correct Answer: f'(g(x)) · g'(x)

Q34. The derivative of inverse sine f(x)=arcsin(x) equals:

• 1/√(1-x²)
• 1/(1-x²)
• 1/√(1+x²)
• 1/(1+x²)

Correct Answer: 1/√(1-x²)

Q35. If velocity v(t)=dx/dt and acceleration a(t)=dv/dt, then a(t) equals:

• d²x/dt
• d x/dt
• d²t/dx
• Integral of v(t)

Correct Answer: d²x/dt

Q36. For f(x)= (2x+1)^-2, f'(x) is:

• -4(2x+1)^-3
• -2(2x+1)^-3
• 4(2x+1)^-3
• (2x+1)^-1

Correct Answer: -4(2x+1)^-3

Q37. If a rate law r=k[A]², derivative dr/d[A] equals:

• 2k[A]
• k[A]²
• 2k[A]²
• k

Correct Answer: 2k[A]

Q38. Which statement about differentiability implies continuity?

• If f is differentiable at a point, then f is continuous at that point
• If f is continuous at a point, then f is differentiable there
• Differentiability and continuity are unrelated
• Continuity implies differentiability for polynomials only

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

Q39. The derivative of f(x)=cos(x) is:

• -sin(x)
• sin(x)
• -cos(x)
• cos(x)

Correct Answer: -sin(x)

Q40. For f(x)=1/x, f'(x) is:

• -1/x²
• 1/x²
• -x
• 0

Correct Answer: -1/x²

Q41. If plasma concentration C(t) has maximum at t*, which condition must hold?

• C'(t*) > 0
• C'(t*) = 0
• C”(t*) > 0 always
• C(t*) = 0

Correct Answer: C'(t*) = 0

Q42. Differentiate f(x)=ln(sin x) using chain rule.

• cos x / sin x
• sin x / cos x
• ln(cos x)
• cos x · ln(sin x)

Correct Answer: cos x / sin x

Q43. The derivative of arccos(x) equals:

• -1/√(1-x²)
• 1/√(1-x²)
• -1/(1-x²)
• 1/(1-x²)

Correct Answer: -1/√(1-x²)

Q44. For f(x)= (x²+1)⁵, f'(x) is:

• 10x(x²+1)⁴
• 5(x²+1)⁴
• 2x(x²+1)⁵
• (x²+1)⁵

Correct Answer: 10x(x²+1)⁴

Q45. If f(x)=u(x)v(x)w(x), derivative requires:

• Pairwise product rule applied three times
• Sum of each function’s derivative times the product of the other two
• Only derivative of the largest function
• Quotient rule

Correct Answer: Sum of each function’s derivative times the product of the other two

Q46. The mean value theorem guarantees a point c where f'(c) equals:

• f(b) – f(a)
• f(b) + f(a)
• (f(b)-f(a))/(b-a)
• 0 always

Correct Answer: (f(b)-f(a))/(b-a)

Q47. The derivative of f(x)=sinh(x) is:

• cosh(x)
• sinh(x)
• e^x
• tanh(x)

Correct Answer: cosh(x)

Q48. If concentration C(t)=At²-Bt+C, at t=1 the derivative C'(1) equals:

• 2A – B
• A – B + C
• 2A + B + C
• A + B

Correct Answer: 2A – B

Q49. For f(x)=arcsin(2x), f'(x) is:

• 2/√(1-4x²)
• 1/√(1-4x²)
• 2/√(1-x²)
• 1/√(1-x²)

Correct Answer: 2/√(1-4x²)

Q50. Which quantity is obtained by differentiating the area function A(t) under a release profile with respect to time?

• Cumulative release
• Instantaneous release rate (flux)
• Total drug released
• Average release over time

Correct Answer: Instantaneous release rate (flux)