Laplace Transform of derivatives MCQs With Answer

The Laplace Transform of derivatives is a core mathematical tool for B. Pharm students studying pharmacokinetics, modeling drug concentration and solving linear differential equations. Understanding properties like L{f’} = sF(s) − f(0), L{f”} = s^2F(s) − s f(0) − f'(0), and transforms involving higher-order derivatives helps in converting time-domain rate equations into algebraic equations in the s-domain. Mastery of initial-condition handling, inverse Laplace techniques, and related properties (shifting, scaling, and differentiation in the s-domain) accelerates problem solving in dosage regimen design and compartmental models. This set of focused Laplace Transform of derivatives MCQs with answers targets common pitfalls and application-based questions for B. Pharm students. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the Laplace transform of f'(t)?

• sF(s) – f(0)
• F'(s)
• 1/s * F(s)
• F(s) – f(0)

Correct Answer: sF(s) – f(0)

Q2. What is the Laplace transform of f”(t)?

• sF(s) – f(0)
• s^2F(s) – s f(0) – f'(0)
• F”(s)
• s^2F(s) – f(0)

Correct Answer: s^2F(s) – s f(0) – f'(0)

Q3. Which property relates multiplication by t in time domain to differentiation in s-domain?

• L{t f(t)} = -dF(s)/ds
• L{t f(t)} = dF(s)/ds
• L{t f(t)} = s dF(s)/ds
• L{t f(t)} = -s dF(s)/ds

Correct Answer: L{t f(t)} = -dF(s)/ds

Q4. If L{f(t)} = F(s), what is L{f”'(t)}?

• s^3 F(s) – s^2 f(0) – s f'(0) – f”(0)
• s^3 F(s) – f(0) – f'(0) – f”(0)

Correct Answer: s^3 F(s) – s^2 f(0) – s f'(0) – f”(0)

Q5. When solving a first-order ODE for drug concentration C'(t) + kC(t) = R(t), using Laplace transforms, what term accounts for the initial concentration C(0)?

• -C(0)
• k C(0)
• s C(0)
• C(0) multiplied by s in transformed equation for C'(t)

Correct Answer: C(0) multiplied by s in transformed equation for C'(t)

Q6. For f(t)=t, what is F(s)=L{t}?

• 1/s
• 1/s^2
• 1/s^3
• 0

Correct Answer: 1/s^2

Q7. Which expression gives L{d^n f / dt^n} in terms of F(s) and initial conditions?

• s^n F(s) – s^{n-1} f(0) – … – f^{(n-1)}(0)
• F^{(n)}(s) + initial terms
• (-1)^n d^n F(s)/ds^n
• s F(s) – f(0)

Correct Answer: s^n F(s) – s^{n-1} f(0) – … – f^{(n-1)}(0)

Q8. How does the Laplace transform help in solving linear ODEs with constant coefficients in pharmacokinetic models?

• Converts differential equations to algebraic equations in s-domain
• Removes need for initial conditions
• Only useful for non-homogeneous equations
• Makes time domain calculations unnecessary but increases complexity

Correct Answer: Converts differential equations to algebraic equations in s-domain

Q9. Given f(0)=2 and L{f’} = sF(s)-2, if L{f’} = 6/(s+1), what is sF(s)-2?

• 6/(s+1)
• 6s/(s+1)
• (6+2s)/(s+1)
• 2/(s+1)

Correct Answer: 6/(s+1)

Q10. Which of the following is the inverse Laplace of 1/(s+a)?

• e^{-at}
• e^{at}
• sin(at)
• cos(at)

Correct Answer: e^{-at}

Q11. In L{f'(t)} = sF(s) – f(0), what role does f(0) play in solving initial value problems?

• Represents initial condition that must be included to get correct solution
• Can be ignored for linear ODEs
• Only needed if f(0) ≠ 0 and time-shifted
• It is absorbed into F(s) automatically

Correct Answer: Represents initial condition that must be included to get correct solution

Q12. If F(s)=1/s and f(0)=0, what is L{f’}?

• 0
• 1
• s*(1/s) – 0 = 1
• 1/s^2

Correct Answer: s*(1/s) – 0 = 1

Q13. Which transform property is useful for handling derivatives of a product t^n f(t)?

• Time multiplication: L{t^n f(t)} = (-1)^n d^n F(s)/ds^n
• Frequency shift property
• Convolution theorem
• Initial value theorem

Correct Answer: Time multiplication: L{t^n f(t)} = (-1)^n d^n F(s)/ds^n

Q14. What is L{δ'(t)} where δ is the Dirac delta? (Assume δ’ denotes derivative)

• s
• 1
• s times L{δ(t)} minus δ(0)
• s * 1 = s

Correct Answer: s

Q15. Which method is commonly combined with Laplace transforms to invert rational F(s) back to f(t)?

• Partial fraction decomposition
• Taylor series expansion
• Numerical integration only
• Fourier transform

Correct Answer: Partial fraction decomposition

Q16. For a second-order linear ODE describing drug kinetics, initial values required for Laplace solution are:

• f(0) and f'(0)
• f(0) only
• f”(0) only
• No initial values

Correct Answer: f(0) and f'(0)

Q17. Which of the following is the Laplace of e^{-at} f(t) (frequency shift)?

• F(s + a)
• F(s – a)
• e^{-a s} F(s)
• F(s) / (s + a)

Correct Answer: F(s + a)

Q18. Given f(t)=cos(ωt), what is L{f'(t)}?

• s * s/(s^2 + ω^2) – 1
• -ω sin(ωt) transformed to -ω^2/(s^2 + ω^2)
• L{-ω sin(ωt)} = -ω * (ω/(s^2+ω^2)) = -ω^2/(s^2+ω^2)
• 0

Correct Answer: L{-ω sin(ωt)} = -ω * (ω/(s^2+ω^2)) = -ω^2/(s^2+ω^2)

Q19. If F(s)=1/(s^2 + 4), what is L{f”(t)} using initial conditions f(0)=0, f'(0)=2?

• s^2 F(s) – s*0 – 2 = s^2/(s^2+4) – 2
• s^2/(s^2+4)
• sF(s)-0
• 2/(s^2+4)

Correct Answer: s^2/(s^2+4) – 2

Q20. Which theorem relates the value of f(0+) to the limit of sF(s) as s → ∞?

• Initial value theorem
• Final value theorem
• Convolution theorem
• Shifting theorem

Correct Answer: Initial value theorem

Q21. What is the Laplace transform of a derivative multiplied by a step u(t-a): L{d/dt [u(t-a) g(t-a)]}?

• e^{-a s} [s G(s) – g(0+)]
• [s G(s) – g(0+)]
• e^{-a s} G(s)
• G'(s)

Correct Answer: e^{-a s} [s G(s) – g(0+)]

Q22. Which expression is true for the derivative of the Laplace transform F(s) with respect to s?

• dF/ds = -L{t f(t)}
• dF/ds = L{t f(t)}
• dF/ds = -L{f'(t)}
• dF/ds = L{f'(t)}

Correct Answer: dF/ds = -L{t f(t)}

Q23. In pharmacokinetics, using Laplace transforms simplifies solving compartment models by:

• Converting time derivatives into algebraic s-terms, incorporating initial concentrations
• Eliminating need to consider elimination rate constants
• Making solutions time-invariant always
• Replacing concentrations with steady-state values only

Correct Answer: Converting time derivatives into algebraic s-terms, incorporating initial concentrations

Q24. What is the Laplace transform of t f'(t) in terms of F(s)?

• -d/ds[sF(s) – f(0)]
• dF/ds
• -dF/ds
• s dF/ds

Correct Answer: -d/ds[sF(s) – f(0)]

Q25. Which approach is correct to invert F(s) that is non-rational (e.g., involves sqrt(s))?

• Use known transform pairs, Bromwich integral or tables, and possibly numerical inversion
• Always use partial fractions
• There is no inverse Laplace
• Differentiate F(s) until it becomes rational

Correct Answer: Use known transform pairs, Bromwich integral or tables, and possibly numerical inversion

Q26. Given L{f(t)} = 3/(s+2), what is L{f'(t)} if f(0)=1?

• s*(3/(s+2)) – 1 = 3s/(s+2) – 1
• 3/(s+2) – 1
• s*(3/(s+2))
• 0

Correct Answer: s*(3/(s+2)) – 1 = 3s/(s+2) – 1

Q27. Which statement about differentiation under the Laplace integral sign is true?

• Differentiating F(s) with respect to s corresponds to multiplication by -t in time domain
• Differentiation with respect to s has no time-domain interpretation
• It corresponds to time derivative f'(t)
• It always simplifies inversion

Correct Answer: Differentiating F(s) with respect to s corresponds to multiplication by -t in time domain

Q28. What is the Laplace transform of the time derivative of a convolution: L{d/dt (f * g)}?

• s F(s) G(s) – (f * g)(0)
• F(s) G(s)
• d/ds [F(s) G(s)]
• 0

Correct Answer: s F(s) G(s) – (f * g)(0)

Q29. Which of these is the correct Laplace transform pair for f'(t) when f(t)=e^{-2t}? (f(0)=1)

• L{f’} = s*(1/(s+2)) – 1 = s/(s+2) – 1
• L{f’} = 1/(s+2)
• L{f’} = -2/(s+2)
• L{f’} = s/(s+2)

Correct Answer: L{f’} = s*(1/(s+2)) – 1 = s/(s+2) – 1

Q30. Which property helps when dealing with derivatives of shifted functions u(t-a)g(t-a)?

• Time-shifting property with multiplication by e^{-as} in s-domain
• Frequency differentiation property
• Convolution with delta function
• Scaling property

Correct Answer: Time-shifting property with multiplication by e^{-as} in s-domain

Q31. Compute L{t^2 f(t)} in terms of F(s).

• L{t^2 f(t)} = d^2 F/ds^2
• L{t^2 f(t)} = (-1)^2 d^2 F/ds^2 = d^2 F/ds^2
• L{t^2 f(t)} = -dF/ds
• L{t^2 f(t)} = s^2 F(s)

Correct Answer: L{t^2 f(t)} = (-1)^2 d^2 F/ds^2 = d^2 F/ds^2

Q32. For a linear system with input R(t) and state variable C(t), Laplace transforms convert C'(t) terms to:

• s C(s) – C(0)
• s C(s)
• C(0)
• C(s)/s

Correct Answer: s C(s) – C(0)

Q33. If L{f(t)} = F(s) and f(0)=0, which of the following is L{∫_0^t f(τ) dτ}?

• F(s)/s
• s F(s)
• 1/F(s)
• F(s) * s

Correct Answer: F(s)/s

Q34. How does one include initial derivative f'(0) when transforming a second-order differential equation?

• Appears as a subtractive term -f'(0) in L{f”}
• Does not appear
• Appears multiplied by s only
• Appears as +f'(0)

Correct Answer: Appears as a subtractive term -f'(0) in L{f”}

Q35. Which Laplace pair is correct for f(t)=t e^{at}?

• L{t e^{at}} = 1/(s-a)^2
• L{t e^{at}} = 1/(s-a)
• L{t e^{at}} = (s-a)^{-3}
• L{t e^{at}} = s/(s-a)^2

Correct Answer: L{t e^{at}} = 1/(s-a)^2

Q36. When using Laplace to solve for concentration, the algebraic equation often requires solving for C(s) as:

• C(s) = (terms from input and initial conditions) / (polynomial in s)
• C(s) = integral of input only
• C(s) = constant only
• C(s) cannot be found using Laplace

Correct Answer: C(s) = (terms from input and initial conditions) / (polynomial in s)

Q37. What is the Laplace transform of cosh(bt)?

• s/(s^2 – b^2)
• s/(s^2 + b^2)
• b/(s^2 – b^2)
• 1/(s – b)

Correct Answer: s/(s^2 – b^2)

Q38. Which of the following is true for stability analysis using Laplace transforms in compartment models?

• Roots of denominator (poles) determine transient behavior and stability
• Only numerator matters
• Laplace cannot analyze stability
• Stability is determined by F(0) only

Correct Answer: Roots of denominator (poles) determine transient behavior and stability

Q39. What is L{f”(t) + 3 f'(t) + 2 f(t)} in terms of F(s) and initial conditions f(0)=a, f'(0)=b?

• (s^2 + 3s + 2) F(s) – s a – b – 3 a
• (s^2 + 3s + 2) F(s)
• F(s) – a – b
• (s^2 + 3s + 2) F(s) – a – b

Correct Answer: (s^2 + 3s + 2) F(s) – s a – b – 3 a

Q40. Inverse Laplace of (s+1)/(s^2 + 4s + 5) corresponds to which time function?

• e^{-2t} cos(t) + e^{-2t} sin(t)
• e^{-t}
• cos(t)
• e^{-2t}

Correct Answer: e^{-2t} cos(t) + e^{-2t} sin(t)

Q41. Which initial/final value theorem statement is correct?

• lim_{t→0+} f(t) = lim_{s→∞} s F(s); lim_{t→∞} f(t) = lim_{s→0} s F(s) if poles permit
• lim_{t→0+} f(t) = lim_{s→0} s F(s)
• Both limits always exist
• These theorems don’t involve sF(s)

Correct Answer: lim_{t→0+} f(t) = lim_{s→∞} s F(s); lim_{t→∞} f(t) = lim_{s→0} s F(s) if poles permit

Q42. If F(s)= (s+3)/(s^2+4s+5), what is the time-domain derivative f'(t) at t=0+ using initial value theorem?

• lim_{s→∞} s[sF(s) – f(0)] gives f'(0+) using transforms of derivative
• Directly equals F(0)
• Cannot be found
• Equal to f(0)

Correct Answer: lim_{s→∞} s[sF(s) – f(0)] gives f'(0+) using transforms of derivative

Q43. Which factor complicates inversion when initial conditions are non-zero?

• Presence of additional polynomial terms from initial conditions in numerator
• They always simplify inversion
• Initial conditions remove poles
• They convert problem into convolution only

Correct Answer: Presence of additional polynomial terms from initial conditions in numerator

Q44. When applying Laplace to systems of ODEs (e.g., two compartments), the transformed algebraic system is solved for:

• Each compartment’s Laplace variable C_i(s) incorporating initial amounts
• Only total mass
• Only steady state values
• Time derivatives directly

Correct Answer: Each compartment’s Laplace variable C_i(s) incorporating initial amounts

Q45. Which of the following is L{sin(ωt)} and useful when differentiating in s-domain?

• ω/(s^2 + ω^2)
• s/(s^2 + ω^2)
• 1/(s + ω)
• 0

Correct Answer: ω/(s^2 + ω^2)

Q46. What is the effect on the Laplace transform of adding a known polynomial initial term (e.g., f(0) and f'(0)) to the numerator?

• Shifts the inverse solution by adding impulses or polynomial terms in time domain
• Removes poles
• No effect on inversion
• Transforms denominator only

Correct Answer: Shifts the inverse solution by adding impulses or polynomial terms in time domain

Q47. How to handle non-zero initial derivatives when using Laplace for pharmacokinetic equations with dosing at t=0?

• Include derivative initial values explicitly in transformed equations L{f’} and L{f”}
• Set them to zero always
• Ignore them if doses are large
• Use Fourier transform instead

Correct Answer: Include derivative initial values explicitly in transformed equations L{f’} and L{f”}

Q48. Which is true for L{d/dt [e^{at} f(t)]} in terms of F(s)?

• Transformed as (s-a) F(s-a) – f(0)
• Equal to s F(s)
• Equal to F(s-a)
• No simple relation exists

Correct Answer: Transformed as (s-a) F(s-a) – f(0)

Q49. Which technique is useful when inverse Laplace yields convolution integrals in time domain?

• Recognize convolution and use convolution theorem to compute time-domain convolution
• Avoid convolution by numerical inversion only
• Convolution never appears for derivatives
• Use only partial fractions

Correct Answer: Recognize convolution and use convolution theorem to compute time-domain convolution

Q50. Which statement best summarizes the utility of Laplace transforms for derivatives in B. Pharm curriculum?

• They convert time-domain differential rate equations with initial conditions into solvable algebraic equations in s-domain, aiding pharmacokinetic modeling
• They only apply to pure mathematics, not pharmacology
• They are obsolete compared to numerical methods
• They remove the role of initial conditions entirely

Correct Answer: They convert time-domain differential rate equations with initial conditions into solvable algebraic equations in s-domain, aiding pharmacokinetic modeling

G S Sachin

I am a Registered Pharmacist under the Pharmacy Act, 1948, and the founder of PharmacyFreak.com. I hold a Bachelor of Pharmacy degree from Rungta College of Pharmaceutical Science and Research. With a strong academic foundation and practical knowledge, I am committed to providing accurate, easy-to-understand content to support pharmacy students and professionals. My aim is to make complex pharmaceutical concepts accessible and useful for real-world application.

Mail- Sachin@pharmacyfreak.com