Derivative of sum/difference of functions MCQs With Answer

Derivative of sum/difference of functions MCQs With Answer is an essential topic for B.Pharm students learning calculus applications in pharmacokinetics and drug formulation. This introduction covers the linearity of differentiation—the sum rule and difference rule—applied to polynomials, exponentials, trigonometric and logarithmic functions commonly seen in rate equations, concentration-time profiles, and reaction kinetics. Practicing these MCQs strengthens skills in simplifying derivatives, evaluating derivatives at specific times, and interpreting physiological rates. Focused practice improves exam performance and practical modeling accuracy in pharmacy problems. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. If f(t)=t^3 and g(t)=2t, what is the derivative of f+g?

• 3t^2 + 2
• t^3 + 2t
• 3t^2
• 6t

Correct Answer: 3t^2 + 2

Q2. Given f(x)=e^(2x) and g(x)=sin x, what is (f-g)’?

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

Correct Answer: 2e^(2x) – cos x

Q3. If C(t)=A e^(-kt) + Bt where A,B,k are constants, C'(t) equals:

• -Ak e^(-kt) + B
• -A k^2 e^(-kt) + B
• A e^(-kt) + B
• -Ak e^(-kt) – B

Correct Answer: -Ak e^(-kt) + B

Q4. True or false: d/dx [u(x)+v(x)] = u'(x) + v'(x).

• True
• False
• Only if u and v are polynomials
• Only if u and v are continuous

Correct Answer: True

Q5. If h(t)=ln t + t^2, h'(t) is:

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

Correct Answer: 1/t + 2t

Q6. For F(x)=3x^4 – 5x^2, F'(x) equals:

• 12x^3 – 10x
• 3*4x^3 – 5*2x^1
• 12x^3 – 10x (same as previous)
• All of the above

Correct Answer: All of the above

Q7. If u(t)=t and v(t)=cos t, derivative of u+v at t=π/2 is:

• 1 – 0 = 1
• 1 + 0 = 1
• 1 – (-1) = 2
• 0

Correct Answer: 1 + 0 = 1

Q8. Which rule explains d/dx[f(x)+g(x)] = f'(x)+g'(x)?

• Linearity (sum rule)
• Product rule
• Chain rule
• Quotient rule

Correct Answer: Linearity (sum rule)

Q9. If R(t)=t^2 + 3t + 5, what is R'(2)?

• 2*2 + 3 = 7
• 4 + 6 + 5 = 15
• 2t + 3 evaluated at 2 = 7
• Both first and third choices

Correct Answer: Both first and third choices

Q10. For S(x)=sin x – x^3, S'(x) is:

• cos x – 3x^2
• cos x + 3x^2
• -sin x – 3x^2
• sin x – 3x^2

Correct Answer: cos x – 3x^2

Q11. If y=f(x)+C where C is constant, dy/dx equals:

• f'(x)
• f'(x)+C
• C
• 0

Correct Answer: f'(x)

Q12. For P(t)=t^5 – e^t, P'(t) equals:

• 5t^4 – e^t
• 5t^4 + e^t
• t^5 – e^t
• 5t^4 – te^(t-1)

Correct Answer: 5t^4 – e^t

Q13. Which is the derivative of f(x)=x^2 + ln x at x=1?

• 2*1 + 1 = 3
• 2 + 0 = 2
• 2 + 1/1 = 3
• 1 + 1 = 2

Correct Answer: 2 + 1/1 = 3

Q14. If M(t)=3sin t – 4cos t, M'(t) is:

• 3cos t + 4sin t
• 3cos t – 4(-sin t)
• 3cos t + 4sin t (equivalent)
• -3cos t + 4sin t

Correct Answer: 3cos t + 4sin t (equivalent)

Q15. Given A(x)=x^2 + x ln x, A'(x) equals:

• 2x + ln x + 1
• 2x + x*(1/x)
• 2x + ln x + 1 (same as previous)
• ln x + 2

Correct Answer: 2x + ln x + 1 (same as previous)

Q16. Evaluate derivative at t=0: D(t)=e^t + t^3.

• e^0 + 0 = 1
• e^0 + 0 = 1 (should be 1 + 0 =1)
• 1 + 0 = 1
• All three are equivalent

Correct Answer: All three are equivalent

Q17. If f(x)=u(x)-v(x) and u’=2x, v’=x^2, then f'(x) is:

• 2x – x^2
• 2x + x^2
• x^2 – 2x
• 0

Correct Answer: 2x – x^2

Q18. For combined functions, d/dx[3x^2 – 4ln x] equals:

• 6x – 4/x
• 3x^2 – 4/x
• 6x – 4ln x
• 6x – 4

Correct Answer: 6x – 4/x

Q19. If concentration C(t)=t + 1/t, what is C'(t)?

• 1 – 1/t^2
• 1 + 1/t^2
• 1 – t
• t – 1/t^2

Correct Answer: 1 – 1/t^2

Q20. Which statement is true regarding linearity of differentiation?

• d/dx[a f(x) + b g(x)] = a f'(x) + b g'(x)
• d/dx[a f(x) + b g(x)] = f'(x) + g'(x)
• Linearity holds only for polynomials
• Linearity fails if a or b are constants

Correct Answer: d/dx[a f(x) + b g(x)] = a f'(x) + b g'(x)

Q21. For Q(x)=x^3 + 4x, Q”(x) using sum/difference rules is:

• 6x
• 3x^2 + 4
• 3x^2 + 4′ = invalid
• 9x^2

Correct Answer: 6x

Q22. If f(t)=sin t + cos t, f'(π) equals:

• cos π – sin π = -1 – 0 = -1
• cos π + -sin π = -1 + 0 = -1
• -1
• All answers are equivalent

Correct Answer: All answers are equivalent

Q23. Identify derivative: y = 2e^x – 5x.

• 2e^x – 5
• e^x – 5x
• 2e^x – 5x
• 2e^x

Correct Answer: 2e^x – 5

Q24. If S(t)=t^2 – 2t + 1/t, S'(1) is:

• 2*1 – 2 – 1 = -1
• 2 – 2 – 1 = -1
• 2 – 2 + (-1) = -1
• All above equivalent

Correct Answer: All above equivalent

Q25. For f(x)=ln x – x^(-1), f'(x) equals:

• 1/x + x^(-2)
• 1/x – (-1)x^(-2)
• 1/x + 1/x^2 (equivalent)
• All three equivalent

Correct Answer: All three equivalent

Q26. In pharmacokinetics, if C(t)=C1(t)+C2(t) where C1 and C2 are compartment concentrations, then C'(t) is:

• C1′(t)+C2′(t)
• C1′(t)-C2′(t)
• Product C1′(t)C2′(t)
• Cannot be determined

Correct Answer: C1′(t)+C2′(t)

Q27. Which derivative is correct for f(x)=x^2 + e^(3x)?

• 2x + 3e^(3x)
• 2x + e^(3x)
• x^2 + 3e^(3x)
• 2x + 3x e^(3x)

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

Q28. If u(x)=sin x and v(x)=ln x, d/dx[u+v] at x=π/2 is:

• cos(π/2) + 1/(π/2) = 0 + 2/π
• 0 + 1 = 1
• 1 + 0 = 1
• cos(π/2) + ln'(π/2)

Correct Answer: cos(π/2) + 1/(π/2) = 0 + 2/π

Q29. If f(x)=3x – 4 and g(x)=x^2, derivative of f-g is:

• 3 – 2x
• 3 – x^2
• 3x – 4 – 2x
• 2x – 3

Correct Answer: 3 – 2x

Q30. For y(t)=5 + 0*t + sin t, y'(t) equals:

• cos t
• 0 + cos t = cos t
• Both first and second choices
• sin t

Correct Answer: Both first and second choices

Q31. If f(x)=x^(1/2) + x^(3/2), f'(x) equals:

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

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

Q32. Which derivative follows from linearity: d/dx[7x^3 – 2e^x] =

• 21x^2 – 2e^x
• 7x^3 – 2e^x
• 21x^3 – 2e^x
• 21x^2 – 2xe^(x-1)

Correct Answer: 21x^2 – 2e^x

Q33. If G(t)=t^2 – ln t – t, G'(t) is:

• 2t – 1/t – 1
• 2t – ln t – 1
• 2t – 1 – 1/t (same as first)
• 2t – t^{-1}

Correct Answer: 2t – 1/t – 1

Q34. For H(x)=cos x + sin x + x, H'(x) equals:

• -sin x + cos x + 1
• -sin x – cos x + 1
• sin x + cos x + 1
• -cos x + sin x + 1

Correct Answer: -sin x + cos x + 1

Q35. If F(t)=e^t + e^(2t), F'(t) is:

• e^t + 2e^(2t)
• e^t + e^(2t)
• e^(t) + 2te^(2t)
• 2e^t + e^(2t)

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

Q36. For J(x)=1/x + x, J'(1) equals:

• -1 + 1 = 0
• -1 + 1 = 0 (units omitted)
• 0
• All are equivalent

Correct Answer: All are equivalent

Q37. If y(x)=ax^n + bx^m where a,b constants, derivative is:

• anx^(n-1) + b m x^(m-1)
• a x^n + b x^m
• n a x^n + m b x^m
• anx^n + bmx^m

Correct Answer: anx^(n-1) + b m x^(m-1)

Q38. Which is derivative of K(t)=ln t – e^t at t=1?

• 1 – e
• 1 – 1 = 0
• 1/e – e
• 1 – e^1 = 1 – e

Correct Answer: 1 – e

Q39. For L(x)=sin x + x^2 + ln x, L'(x) simplifies to:

• cos x + 2x + 1/x
• cos x + 2x
• sin x + 2x + 1/x
• cos x + x + 1/x

Correct Answer: cos x + 2x + 1/x

Q40. If total rate R(t)=r1(t) – r2(t) and r1′(t)=2, r2′(t)=3, then R'(t) is:

• -1
• 5
• -5
• 1

Correct Answer: -1

Q41. When differentiating sum of many terms, which is true?

• Differentiate term-by-term and add results
• Must combine terms first before differentiating
• Only first and last terms differentiate
• Cannot differentiate sums

Correct Answer: Differentiate term-by-term and add results

Q42. If f(x)=x^3 + 4x – 7, f'(x) equals:

• 3x^2 + 4
• 3x^3 + 4
• 3x^2 + 4x
• x^2 + 4

Correct Answer: 3x^2 + 4

Q43. For m(t)=t^4 – 4t^2 + 2, m'(t) is:

• 4t^3 – 8t
• t^4 – 8t
• 4t^3 – 8t + 2
• 4t^3 – 8

Correct Answer: 4t^3 – 8t

Q44. If y = e^(at) + e^(bt), dy/dt is:

• a e^(at) + b e^(bt)
• a e^(at) – b e^(bt)
• e^(at) + e^(bt)
• ab e^(a+b)t

Correct Answer: a e^(at) + b e^(bt)

Q45. Consider N(x)=sqrt(x) + 1/x, N'(x) equals:

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

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

Q46. If rate functions r(t)=t^2 and s(t)=t^3, derivative of r+s at t=2 is:

• 2*2 + 3*(2^2) = 4 + 12 = 16
• 4 + 12 = 16 (equivalent)
• Both above
• Only 16 if units match

Correct Answer: Both above

Q47. For O(x)=x ln x – x, O'(x) simplifies to:

• ln x + 1 – 1 = ln x
• ln x + 1
• ln x
• 1

Correct Answer: ln x

Q48. If S(x)=f(x)+g(x)+h(x), what is S'(x)?

• f'(x)+g'(x)+h'(x)
• f'(x)g'(x)h'(x)
• f(x)+g'(x)+h'(x)
• Cannot be determined

Correct Answer: f'(x)+g'(x)+h'(x)

Q49. Given f(x)=5 and g(x)=x^2, derivative of f+g is:

• 0 + 2x = 2x
• 5 + 2x
• 2x
• Both first and third choices

Correct Answer: Both first and third choices

Q50. In practice, why is the sum/difference rule important for B.Pharm students?

• Enables term-by-term differentiation in kinetics and dosing models
• Only theoretical interest, not used in pharmacy
• Replaces need for product and chain rules
• Allows integration without limits

Correct Answer: Enables term-by-term differentiation in kinetics and dosing models

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