Successive Differentiation MCQs With Answer

Successive Differentiation MCQs With Answer are essential practice for B. Pharm students to master higher order derivatives, Leibniz rule, and nth-derivative formulas used in pharmacokinetics, drug dissolution, and dosage-response modeling. This concise, Student-friendly post focuses on successive differentiation techniques, common patterns (polynomials, exponentials, trigonometric functions), and applications in pharmaceutical calculations. Practicing these MCQs sharpens analytic skills for exam preparation and real-world problems like reaction-rate kinetics and concentration–time curves. Each question emphasizes conceptual clarity and computational accuracy so students can identify patterns for quick successive differentiation. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the second derivative d2/dx2 of f(x) = x^3 ?

• 6x
• 3x^2
• 2x
• 9x^2

Correct Answer: 6x

Q2. If f(x) = e^{2x}, what is the second derivative f”(x)?

• 4e^{2x}
• 2e^{2x}
• e^{2x}
• 8e^{2x}

Correct Answer: 4e^{2x}

Q3. For f(x) = sin x, what is f”(x)?

• sin x
• -sin x
• cos x
• -cos x

Correct Answer: -sin x

Q4. The nth derivative of x^n is:

• n! (constant)
• 0
• n x^{n-1}
• (n-1)! x

Correct Answer: n! (constant)

Q5. Using Leibniz rule, the second derivative of u(x)v(x) is:

• u”v + 2u’v’ + uv”
• u”v + u’v’ + uv”
• u”v – 2u’v’ + uv”
• u’v + uv’

Correct Answer: u”v + 2u’v’ + uv”

Q6. If f(x)=ln x, what is f”(x) for x>0?

• 1/x^2
• -1/x^2
• -1/x
• 1/x

Correct Answer: -1/x^2

Q7. For f(x)=cos x, what is the third derivative f”'(x)?

• -cos x
• sin x
• -sin x
• cos x

Correct Answer: sin x

Q8. The first derivative of y = x^{-2} is:

• -2x^{-3}
• -x^{-3}
• 2x^{-1}
• 2x^{-3}

Correct Answer: -2x^{-3}

Q9. If C(t)=C_0 e^{-kt} (pharmacokinetic decay), what is d^2C/dt^2 ?

• k^2 C_0 e^{-kt}
• k C_0 e^{-kt}
• -k C_0 e^{-kt}
• k^2 C_0 e^{kt}

Correct Answer: k^2 C_0 e^{-kt}

Q10. The third derivative of f(x)=x^4 is:

• 24x
• 12x^2
• 24
• 4x^3

Correct Answer: 24x

Q11. Which is true for successive differentiation of e^{ax} ?

• The nth derivative is a^n e^{ax}
• The nth derivative is n a e^{ax}
• The nth derivative is e^{ax}/a^n
• It becomes zero for n>a

Correct Answer: The nth derivative is a^n e^{ax}

Q12. For f(x)=sin(2x), f”(x) equals:

• -4 sin(2x)
• -2 sin(2x)
• -sin(2x)
• 4 sin(2x)

Correct Answer: -4 sin(2x)

Q13. If y = x e^{x}, y” is:

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

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

Q14. The second derivative of f(x)=arctan x is:

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

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

Q15. If f(x)=ln(1+x), the second derivative at x=0 equals:

• -1
• 1
• 0
• 2

Correct Answer: -1

Q16. The fourth derivative of sin x is:

• sin x
• -sin x
• cos x
• -cos x

Correct Answer: sin x

Q17. For f(x)=x^5, f^{(3)}(x) equals:

• 60x^2
• 20x^3
• 120x
• 60x^3

Correct Answer: 60x^2

Q18. Using successive differentiation, derivative of product uv for third order involves how many terms per Leibniz rule?

• 4 terms
• 3 terms
• 2 terms
• 5 terms

Correct Answer: 4 terms

Q19. The second derivative of y = 1/(1+x) is:

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

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

Q20. If f(x)=x^2 sin x, what is f”(x) at x=0 ?

• 0
• 2
• 1
• -1

Correct Answer: 0

Q21. The nth derivative of cos(bx) is:

• b^n cos(bx + nπ/2)
• b^n cos(bx)
• b^n sin(bx)
• n b cos(bx)

Correct Answer: b^n cos(bx + nπ/2)

Q22. For f(x)=e^{ax} cos(bx), successive derivatives combine which functions?

• e^{ax} cos(bx) and e^{ax} sin(bx)
• only e^{ax} cos(bx)
• only polynomials
• logarithms and exponentials

Correct Answer: e^{ax} cos(bx) and e^{ax} sin(bx)

Q23. The third derivative of f(x)=e^{3x} is:

• 27 e^{3x}
• 9 e^{3x}
• 3 e^{3x}
• 81 e^{3x}

Correct Answer: 27 e^{3x}

Q24. If rate r(t)=kt^2, the second derivative d^2r/dt^2 equals:

• 2k
• k
• 4kt
• 0

Correct Answer: 2k

Q25. Successive differentiation of polynomial eventually gives:

• 0 (after finite steps)
• an exponential
• a sine function
• a logarithm

Correct Answer: 0 (after finite steps)

Q26. The second derivative of f(x)=t^n (with respect to t) is:

• n(n-1) t^{n-2}
• n t^{n-1}
• n^2 t^{n-2}
• (n-1) t^{n-2}

Correct Answer: n(n-1) t^{n-2}

Q27. For f(x)=x ln x, f”(x) equals:

• 1/x
• ln x
• 1
• 0

Correct Answer: 1/x

Q28. The second derivative of f(x)=cosh x is:

• cosh x
• sinh x
• -cosh x
• -sinh x

Correct Answer: cosh x

Q29. The nth derivative of a constant is:

• 0 for n>=1
• the constant
• n times the constant
• infinite

Correct Answer: 0 for n>=1

Q30. If f(x)=tan x, f”(x) can be expressed in terms of:

• tan x and sec^2 x
• only tan x
• only sec x
• only cos x

Correct Answer: tan x and sec^2 x

Q31. For successive differentiation, which tool gives a direct formula for derivatives of product of two functions?

• Leibniz rule
• Integration by parts
• L’Hôpital’s rule
• Mean value theorem

Correct Answer: Leibniz rule

Q32. The second derivative of f(x)=1/x^2 is:

• 6/x^4
• -6/x^4
• 2/x^3
• -2/x^3

Correct Answer: 6/x^4

Q33. If y = (ax + b)^n, the second derivative y” equals:

• n(n-1)a^2 (ax+b)^{n-2}
• n a (ax+b)^{n-1}
• a^2 (ax+b)^n
• n(n-1) (ax+b)^{n}

Correct Answer: n(n-1)a^2 (ax+b)^{n-2}

Q34. Successive differentiation of sin^2 x yields periodic derivatives. The second derivative of sin^2 x is:

• 2 cos 2x
• -2 cos 2x
• 2 sin 2x
• -2 sin 2x

Correct Answer: 2 cos 2x

Q35. If f(x)=x e^{-x}, f”(x) equals:

• e^{-x}(x – 2)
• e^{-x}(x + 2)
• -e^{-x}(x + 2)
• e^{-x} x

Correct Answer: e^{-x}(x – 2)

Q36. The third derivative of f(x)=ln x at x=1 is:

• 2
• -2
• -1
• 1

Correct Answer: 2

Q37. For successive differentiation, the derivative of inverse function y=f^{-1}(x) uses:

• 1/(f'(y))
• f'(x)
• f”(x)
• 0

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

Q38. The second derivative of f(x)=sinh(ax) is:

• a^2 sinh(ax)
• a sinh(ax)
• a^2 cosh(ax)
• 0

Correct Answer: a^2 sinh(ax)

Q39. If concentration C(t)=At^3, the third derivative d^3C/dt^3 equals:

• 6A
• 3A
• 0
• tA

Correct Answer: 6A

Q40. The nth derivative of sin x evaluated at x=0 cycles with period:

• 4
• 2
• π
• 3

Correct Answer: 4

Q41. For f(x)=e^{x} x^2, successive derivatives will be combinations of:

• e^{x} times polynomials
• only polynomials
• only exponentials
• trig functions

Correct Answer: e^{x} times polynomials

Q42. The second derivative of f(x)=sec x is:

• sec x (tan^2 x + sec^2 x)
• sec x tan x
• sec x
• tan^2 x

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

Q43. If y = (sin x)/x, the first derivative at x->0 (use limit) is:

• 0
• 1
• -1/3
• undefined

Correct Answer: 0

Q44. The third derivative of f(x)=x^3 ln x is:

• 6 ln x + 11
• 6 ln x + 6
• 6 ln x + 3
• 6

Correct Answer: 6 ln x + 11

Q45. For f(x)=e^{ax} sin(bx), the first derivative includes which factors?

• a e^{ax} sin(bx) + b e^{ax} cos(bx)
• only a e^{ax} sin(bx)
• only b e^{ax} cos(bx)
• e^{ax} (a+b) sin(bx)

Correct Answer: a e^{ax} sin(bx) + b e^{ax} cos(bx)

Q46. The second derivative of f(x)=log_a x equals (natural log used):

• -1/(x^2 (ln a))
• 1/(x ln a)
• -1/(x ln a)
• 1/(x^2 ln a)

Correct Answer: -1/(x^2 (ln a))

Q47. Successive differentiation helps find Taylor series coefficients because coefficients relate to:

• nth derivatives at a point
• integrals
• limits at infinity
• only first derivative

Correct Answer: nth derivatives at a point

Q48. If f(x)=x^n e^{x}, the nth derivative has leading term:

• n! e^{x}
• e^{x} x^n
• x^n
• n e^{x}

Correct Answer: n! e^{x}

Q49. The second derivative test for extrema uses successive differentiation to evaluate:

• concavity via f”(x)
• only f'(x)
• inflection via f^{(3)}(x)
• integral of f(x)

Correct Answer: concavity via f”(x)

Q50. The general Leibniz formula gives the nth derivative of product u*v as a sum of:

• binomial-weighted derivatives of u and v
• a single term
• only u^{(n)} v
• only v^{(n)} u

Correct Answer: binomial-weighted derivatives of u and v

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