Derivative of product of two functions MCQs With Answer

The derivative of product of two functions MCQs With Answer

The derivative of the product of two functions is essential for B.Pharm students who apply calculus to pharmacokinetics, dosage modeling and drug interaction rates. Understanding the product rule (d(uv)/dx = u’v + uv’) helps you differentiate combined polynomial, exponential, trigonometric and logarithmic expressions commonly found in concentration-time equations. These MCQs emphasize conceptual clarity, step-by-step differentiation, common pitfalls, and real pharmacy applications such as rate of change in drug concentration and bioavailability models. Practice strengthens problem-solving speed and accuracy for exams and research calculations. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. If u(x)=x^2 and v(x)=e^x, what is d(uv)/dx?

• e^x(2x + x^2)
• 2x e^x
• x^2 e^x
• e^x(2 + x)

Correct Answer: e^x(2x + x^2)

Q2. If u(x)=sin x and v(x)=cos x, what is d(uv)/dx?

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

Correct Answer: cos^2 x – sin^2 x

Q3. For u(x)=ln x and v(x)=x^3, find d(uv)/dx.

• 3x^2 ln x + x^2
• x^2(1 + 3 ln x)
• 3x^2 + ln x
• x^3/x + 3x^2

Correct Answer: x^2(1 + 3 ln x)

Q4. If y=5·u(x)·v(x), how does the constant affect the derivative?

• y’ = 5(u’v’ )
• y’ = 5(u’v + uv’)
• y’ = u’v + uv’
• y’ = 5u’v

Correct Answer: y’ = 5(u’v + uv’)

Q5. Given u(x)=7 (constant) and v(x)=x^4, what is d(uv)/dx?

• 28x^3
• 7x^4
• 4x^3
• 0

Correct Answer: 28x^3

Q6. If f(x)=x sin x, then f'(x) equals?

• cos x + x sin x
• sin x + x cos x
• cos x – x sin x
• sin x – x cos x

Correct Answer: sin x + x cos x

Q7. For f(x)=x^2 e^{2x}, what is f'(x)?

• e^{2x}(2x + 2x^2)
• 2x e^{2x} + 2x^2 e^{2x}
• 2e^{2x}x(1 + x)
• All of the above

Correct Answer: All of the above

Q8. What is the standard product rule formula?

• (uv)’ = u’v’ + uv
• (uv)’ = u’v + uv’
• (uv)’ = u’v – uv’
• (uv)’ = uv’

Correct Answer: (uv)’ = u’v + uv’

Q9. If f(x)=x^3 ln(x^2), what is f'(x)?

• 3x^2 ln(x^2) + 2x^2
• x^3·(2/x) + ln(x^2)
• 6x ln x + 3x^2
• 3x^2 ln x + 2x

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

Q10. Differentiate f(x)=e^x sin x.

• e^x(sin x – cos x)
• e^x(cos x + sin x)
• e^x cos x
• e^x sin x

Correct Answer: e^x(cos x + sin x)

Q11. If C(t)=A(t)·B(t) and at t0: A=3, A’=-0.5, B=4, B’=0.2, what is C'(t0)?

• -0.5·4 + 3·0.2
• 3·4 + (-0.5)·0.2
• -2 + 0.6 = -1.4
• 1.0

Correct Answer: -2 + 0.6 = -1.4

Q12. If u and v are nonzero and (uv)’ = 0, which condition is correct?

• u’v + uv’ = 0
• u’ = v’
• u’ = 0 and v’ = 0
• u = v

Correct Answer: u’v + uv’ = 0

Q13. If y=(sin x)(ln x), y’ equals?

• cos x ln x + sin x / x
• ln x cos x + 1/x
• cos x / x + sin x ln x
• sin x cos x + ln x

Correct Answer: cos x ln x + sin x / x

Q14. For f(x)=(tan x)(x^2), what is f'(x)?

• x^2 sec^2 x + 2x tan x
• 2x tan x + x^2 tan’ x
• 2x tan x + x^2 sec^2 x
• x^2 sec x + 2x tan x

Correct Answer: x^2 sec^2 x + 2x tan x

Q15. If u(x)=x e^{-x} and v(x)=cos x, compute (uv)’.

• e^{-x}cos x – x e^{-x} cos x – x e^{-x} sin x
• e^{-x}cos x + x(-e^{-x})cos x + x e^{-x}(-sin x)
• e^{-x}cos x – x e^{-x}sin x
• e^{-x}(cos x – x sin x) – x e^{-x} cos x

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

Q16. Which expression correctly simplifies d/dx[x·ln x]?

• ln x + 1
• x·(1/x) + ln x
• ln x + x/x
• All of the above

Correct Answer: All of the above

Q17. Let f(x)= (x^2 + 1)(x^3 – x). What is f'(x)?

• (2x)(x^3 – x) + (x^2 + 1)(3x^2 – 1)
• 2x(x^3 – x) + (x^2 + 1)(3x^2 – 1)
• (2x + 3x^2 – 1)(x^3 – x + x^2 + 1)
• 5x^4 – x^3 + 2x

Correct Answer: (2x)(x^3 – x) + (x^2 + 1)(3x^2 – 1)

Q18. If u(x)=x^n and v(x)=x^m, derivative of uv is:

• n x^{n-1} x^m + x^n m x^{m-1}
• (n+m) x^{m+n-1}
• x^{m+n}(n+m)
• n m x^{n+m-2}

Correct Answer: n x^{n-1} x^m + x^n m x^{m-1}

Q19. For h(x) = (e^{x^2})(\sin x), h'(x) equals?

• 2x e^{x^2} sin x + e^{x^2} cos x
• e^{x^2}(2x sin x + cos x)
• Both A and B
• e^{x^2} sin x + e^{x^2} cos x

Correct Answer: Both A and B

Q20. Which is true for the derivative of product u·v when u = v?

• (u^2)’ = 2u u’
• (u^2)’ = u’ u’
• (u^2)’ = 2u’ u’
• (u^2)’ = u^2′

Correct Answer: (u^2)’ = 2u u’

Q21. If A(t)=t^2 and B(t)=1/t, what is d/dt[A·B] for t ≠ 0?

• 2t·(1/t) + t^2·(-1/t^2)
• 2 + (-1)
• 1
• 2t/t – t^2/t^2

Correct Answer: 2t·(1/t) + t^2·(-1/t^2)

Q22. Differentiate f(x)=(ln x)(e^{x}).

• (1/x)e^{x} + ln x e^{x}
• e^{x}/x + e^{x} ln x
• e^{x}(1/x + ln x)
• All of the above

Correct Answer: All of the above

Q23. For y = x·cos x, y’ equals?

• cos x – x sin x
• cos x + x sin x
• sin x – x cos x
• -sin x + x cos x

Correct Answer: cos x – x sin x

Q24. If f(x)=(ax)(e^{bx}) with constants a,b, derivative is:

• a e^{bx} + abx e^{bx}
• ae^{bx}(1 + bx)
• Both A and B
• ab e^{bx}x

Correct Answer: Both A and B

Q25. Which option gives d/dx[(sin x)(cos x)]?

• cos^2 x – sin^2 x
• 2 cos x (-sin x)
• sin 2x
• 2 sin x cos x

Correct Answer: cos^2 x – sin^2 x

Q26. If f(x)=x·ln(1+x), what is f'(x)?

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

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

Q27. Suppose R(t)=P(t)·Q(t) and P(t)=e^{2t}, Q(t)=t^3. R'(t) is:

• 2e^{2t}·t^3 + e^{2t}·3t^2
• e^{2t}(2t^3 + 3t^2)
• Both A and B
• e^{2t}t^2(2t + 3)

Correct Answer: Both A and B

Q28. For f(x) = (1/x)(sin x), find f'(x).

• (-1/x^2) sin x + (1/x) cos x
• cos x / x – sin x / x^2
• (cos x x – sin x)/x^2
• All of the above

Correct Answer: All of the above

Q29. If u(x)=arctan x and v(x)=x, then (uv)’ is:

• 1·arctan x + x·(1/(1+x^2))
• arctan x + x/(1+x^2)
• Both A and B
• x/(1+x^2)

Correct Answer: Both A and B

Q30. Determine derivative of f(x)=x e^{x} sin x using product rule iteratively.

• e^{x} sin x + x e^{x} sin x + x e^{x} cos x
• e^{x} sin x + x e^{x}(sin x + cos x)
• e^{x}[sin x + x(sin x + cos x)]
• All are equivalent

Correct Answer: All are equivalent

Q31. If F(x)= (x^2)(cos x)(e^x), which strategy is best?

• Differentiate as product of two groups: (x^2)(cos x e^x) applying product rule twice
• Differentiate cos x and e^x together using chain rule only
• Convert to sum of logs first then differentiate
• Use quotient rule

Correct Answer: Differentiate as product of two groups: (x^2)(cos x e^x) applying product rule twice

Q32. What is d/dx[(x+1)(ln x – 1)]?

• (1)(ln x -1) + (x+1)(1/x)
• ln x -1 + (x+1)/x
• ln x -1 + 1 + 1/x
• Both A and B

Correct Answer: Both A and B

Q33. If y = u(x)v(x) and u = e^{g(x)}, which rule(s) apply?

• Product rule only
• Chain rule only
• Both product and chain rules
• Quotient rule

Correct Answer: Both product and chain rules

Q34. For f(x)=x·(1 + x)^{5}, f'(x) equals?

• (1)(1+x)^5 + x·5(1+x)^4
• (1+x)^4[(1+x) + 5x]
• (1+x)^4(1 + 6x)
• All of the above

Correct Answer: All of the above

Q35. If concentration C(t)=k t e^{-kt} (k constant), C'(t) is:

• k e^{-kt} + k t (-k) e^{-kt}
• k e^{-kt}(1 – kt)
• Both A and B
• -k^2 t e^{-kt}

Correct Answer: Both A and B

Q36. Which is derivative of f(x)= (sin x)(ln x^2) ?

• cos x ln x^2 + sin x·(2/x)
• ln x^2 cos x + 2 sin x / x
• Both A and B
• cos x ln x + sin x / x

Correct Answer: Both A and B

Q37. If u(x)=x^2 and v(x)=sin(x^2), compute d(uv)/dx.

• 2x sin(x^2) + x^2 cos(x^2)·2x
• 2x sin(x^2) + 2x^3 cos(x^2)
• 2x[sin(x^2) + x^2 cos(x^2)]
• All of the above

Correct Answer: All of the above

Q38. For product of three functions u·v·w, the derivative is best obtained by:

• Applying product rule to u and (v·w)
• Using triple product formula directly as u’vw + uv’w + uvw’
• Either A or B
• Using quotient rule

Correct Answer: Either A or B

Q39. If f(x)=x·cosh x, what is f'(x)? (cosh’ = sinh)

• cosh x + x sinh x
• sinh x + x cosh x
• cosh x – x sinh x
• x cosh x

Correct Answer: cosh x + x sinh x

Q40. Given g(x) = (ln x)(1/x), g'(x) equals?

• (1/x)(1/x) + ln x (-1/x^2)
• 1/x^2 – ln x / x^2
• (1 – ln x)/x^2
• All of the above

Correct Answer: All of the above

Q41. If a drug concentration model is C(t)=t·e^{-0.1t}, what is dC/dt?

• e^{-0.1t} – 0.1t e^{-0.1t}
• e^{-0.1t}(1 – 0.1t)
• Both A and B
• -0.1 e^{-0.1t}

Correct Answer: Both A and B

Q42. Which derivative rule combination is needed for f(x)=(sin(x^2))(e^{3x})?

• Product rule and chain rule
• Only product rule
• Only chain rule
• Quotient rule

Correct Answer: Product rule and chain rule

Q43. If y=(x^3)(arcsin x), what is y’ (conceptually)?

• 3x^2 arcsin x + x^3/(sqrt(1-x^2))
• arcsin x + x^3/(sqrt(1-x^2))
• x^3 arcsin’ x
• 3x^2 arcsin x

Correct Answer: 3x^2 arcsin x + x^3/(sqrt(1-x^2))

Q44. Find derivative of f(x)=(cos x)(ln(1+x)).

• -sin x ln(1+x) + cos x /(1+x)
• cos x /(1+x) – sin x ln(1+x)
• Both A and B
• -ln(1+x) cos x

Correct Answer: Both A and B

Q45. If u(x)=x and v(x)=x·e^x, what is (uv)’ simplified?

• 1·(x e^x) + x·(e^x + x e^x)
• x e^x + x e^x + x^2 e^x
• e^x x (2 + x)
• All of the above

Correct Answer: All of the above

Q46. When using the product rule with implicit differentiation for u(x,y)·v(x,y)=0, you should:

• Differentiation treat y as function of x and apply product rule
• Ignore y’ terms
• Use partial derivatives only
• Rearrange to v = 0 first

Correct Answer: Differentiation treat y as function of x and apply product rule

Q47. If f(x)=(x^2 + x)(e^x), f'(x) equals?

• (2x + 1)e^x + (x^2 + x)e^x
• e^x(2x + 1 + x^2 + x)
• e^x(x^2 + 3x + 1)
• All of the above

Correct Answer: All of the above

Q48. For u(t)=t^4 and v(t)=cos(2t), u’v + uv’ simplifies to:

• 4t^3 cos(2t) – t^4·2 sin(2t)
• 4t^3 cos(2t) – 2t^4 sin(2t)
• 4t^3 cos(2t) – 2t^4 sin(2t) (same as A)
• All options are equivalent

Correct Answer: All options are equivalent

Q49. If f(x)= (sin x)^2 · x, what is f'(x)?

• 2 sin x cos x · x + (sin x)^2
• x sin 2x + sin^2 x
• Both A and B
• sin^2 x

Correct Answer: Both A and B

Q50. Which statement best describes the role of the product rule in pharmacokinetics?

• It helps differentiate models where concentration is product of time-dependent factors
• It is irrelevant; only integrals are used in pharmacokinetics
• It replaces the need for chain rule entirely
• It only applies to constant functions

Correct Answer: It helps differentiate models where concentration is product of time-dependent factors

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