Derivative of ex MCQs With Answer

Introduction: The “Derivative of ex MCQs With Answer” collection is designed specifically for B. Pharm students to master exponential differentiation and its pharmaceutical applications. This Student-friendly guide covers derivative rules for e^x, e^{kx}, chain rule, product and quotient with exponential terms, and pharmacokinetic links such as first-order decay, rate constants and half-life calculations. Each question reinforces conceptual understanding and calculation skills needed for drug kinetics, formulation science, and analytical problems. Clear explanations and varied problem types help you apply exponential derivatives to real B. Pharm scenarios. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the derivative of e^x with respect to x?

• e^x
• 1
• x e^{x-1}
• ln e

Correct Answer: e^x

Q2. What is the derivative of e^{kx} where k is a constant?

• k e^{kx}
• e^{kx}/k
• k x e^{kx-1}
• e^{xk^2}

Correct Answer: k e^{kx}

Q3. Which expression is the derivative of a^x (a > 0, a ≠ 1)?

• a^x ln a
• a^{x-1}
• x a^{x-1}
• e^{x ln a}/x

Correct Answer: a^x ln a

Q4. Using the chain rule, what is d/dx [e^{f(x)}]?

• f'(x) e^{f(x)}
• e^{f'(x)}
• f(x) e^{f(x)-1}
• ln f(x) e^{f(x)}

Correct Answer: f'(x) e^{f(x)}

Q5. What is the derivative of x e^x?

• e^x (x + 1)
• e^x x
• e^{x+1}
• 1 + e^x

Correct Answer: e^x (x + 1)

Q6. Compute d/dx [e^{2x^2}].

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

Correct Answer: 4x e^{2x^2}

Q7. What is d/dx [e^{-kt}] with respect to t (k constant)?

• -k e^{-kt}
• k e^{-kt}
• e^{-kt}/k
• 0

Correct Answer: -k e^{-kt}

Q8. If y = e^{x}/x, what is dy/dx?

• e^x (x – 1)/x^2
• e^x/x
• e^x (x + 1)/x^2
• e^{x-1}/x

Correct Answer: e^x (x – 1)/x^2

Q9. What is the second derivative d^2/dx^2 [e^{ax}] (a constant)?

• a^2 e^{ax}
• a e^{ax}
• e^{ax}
• a^3 e^{ax}

Correct Answer: a^2 e^{ax}

Q10. d/dx [ln(e^x)] equals:

• 1
• e^x
• ln e
• 0

Correct Answer: 1

Q11. For f(x)=e^{\sin x}, what is f'(x)?

• \u03bcos x e^{\sin x}
• \u03bsin x e^{\cos x}
• e^{\sin x}
• \u03bcos x + e^{\sin x}

Correct Answer: cos x e^{sin x}

Q12. Which statement about derivatives of e^{x} is true?

• All derivatives of e^{x} equal e^{x}
• Only the first derivative equals e^{x}
• The derivative becomes polynomial times e^{x}
• Derivatives alternate signs

Correct Answer: All derivatives of e^{x} equal e^{x}

Q13. In pharmacokinetics, if C(t)=C_0 e^{-kt}, what is dC/dt?

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

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

Q14. If concentration follows first-order decay C=C_0 e^{-kt}, the slope of ln C vs t equals:

• -k
• k
• ln C_0
• C_0

Correct Answer: -k

Q15. The half-life t_{1/2} for a first-order process is given by:

• ln 2 / k
• k / ln 2
• 2 / k
• ln k / 2

Correct Answer: ln 2 / k

Q16. If ln C vs t has slope -0.2 h^{-1}, what is the half-life (h)?

• ln 2 / 0.2 ≈ 3.465 h
• 0.2 / ln 2 ≈ 0.289 h
• ln 2 × 0.2 ≈ 0.139 h
• 2 / 0.2 = 10 h

Correct Answer: ln 2 / 0.2 ≈ 3.465 h

Q17. d/dx [e^{x^2}] evaluated at x=1 is:

• 2e
• e
• 0
• 1

Correct Answer: 2e

Q18. d/dx [e^{3x+2}] equals:

• 3 e^{3x+2}
• e^{3x+2}
• (3x+2) e^{3x+2}
• e^{3x+2}/3

Correct Answer: 3 e^{3x+2}

Q19. d/dx [e^{\ln x}] for x>0 is:

• 1
• e^{\ln x} / x
• ln x
• 1/x

Correct Answer: 1

Q20. Which is the derivative of ln(1+e^x)?

• e^x / (1 + e^x)
• 1 / (1 + e^x)
• ln(1+e^x) / e^x
• e^{-x} / (1 + e^{-x})

Correct Answer: e^x / (1 + e^x)

Q21. d/dx [e^x/(1+e^x)] equals:

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

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

Q22. If f(x)=x^2 e^x, what is f'(x)?

• e^x (x^2 + 2x)
• 2x e^{x-1}
• x^2 e^{x-1}
• 2 e^x x^2

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

Q23. d/dx [e^{x} \u2212 \u03bcos x] equals:

• e^x + sin x
• e^x – sin x
• e^x + cos x
• -e^x + sin x

Correct Answer: e^x + sin x

Q24. Which derivative gives the slope of the tangent to y=e^x at x=0?

• 1
• e^0 = 1
• Both 1 and e^0 = 1
• 0

Correct Answer: Both 1 and e^0 = 1

Q25. d/dx [e^{x} \u00b7 \u221as x] (product) is:

• e^x \u221a x + e^x/(2 \u221a x)
• e^x \u221a x – e^x/(2 \u221a x)
• \u221a x
• e^x/(2 \u221a x)

Correct Answer: e^x √x + e^x/(2 √x)

Q26. d/dx [e^{g(x)} h(x)] equals (product rule):

• h'(x) e^{g(x)} + h(x) g'(x) e^{g(x)}
• h(x) e^{g'(x)}
• g'(x) h'(x) e^{g(x)}
• e^{g(x)} (h(x) + g(x))

Correct Answer: h'(x) e^{g(x)} + h(x) g'(x) e^{g(x)}

Q27. Evaluate d/dx [e^{x} ln x] for x>0.

• e^x ln x + e^x/x
• e^x / ln x
• ln x / e^x
• e^x ln x – e^x/x

Correct Answer: e^x ln x + e^x/x

Q28. For y=e^{u(x)} and u(x)=ln x^2, what is dy/dx?

• (2/x) e^{ln x^2}
• e^{ln x^2}/(2x)
• 2x e^{ln x^2}
• e^{ln x^2}

Correct Answer: (2/x) e^{ln x^2}

Q29. Simplify (2/x) e^{ln x^2} for x>0.

• 2x/x = 2
• 2x
• 2/x
• 2 x^2

Correct Answer: 2/x

Q30. d/dx [e^{x^3}] is:

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

Correct Answer: 3x^2 e^{x^3}

Q31. If f(x)=e^{x} and g(x)=x, what is d/dx [f(g(x))]?

• g'(x) e^{g(x)} = e^x
• f'(x) g'(x) = x e^x
• e^{x} x’
• 0

Correct Answer: g'(x) e^{g(x)} = e^x

Q32. d/dx [e^{x} + e^{-x}] equals:

• e^x – e^{-x}
• e^x + e^{-x}
• 0
• 2 e^x

Correct Answer: e^x – e^{-x}

Q33. If y = e^{ax} / x where a is constant, dy/dx is:

• (a x – 1) e^{ax} / x^2
• a e^{ax} / x
• e^{ax} / x^2
• (a – 1) e^{ax} / x

Correct Answer: (a x – 1) e^{ax} / x^2

Q34. For small h, e^{x+h} \u2212 e^x ≈:

• h e^x
• e^h
• e^x / h
• h / e^x

Correct Answer: h e^x

Q35. The Maclaurin series derivative property: derivative of sum \u2211 e^x/n! equals:

• e^x (same series shifted)
• 0
• 1
• Sum of reciprocals

Correct Answer: e^x (same series shifted)

Q36. If rate of elimination r(t)=k e^{-kt}, what is dr/dt?

• -k^2 e^{-kt}
• k^2 e^{-kt}
• -k e^{-kt}
• 0

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

Q37. Which technique is used to differentiate e^{x} \u00b7 \u221as x when x>0?

• Product rule plus chain rule
• Quotient rule only
• Integration by parts
• Logarithmic differentiation only

Correct Answer: Product rule plus chain rule

Q38. d/dx [e^{x} \u2212 x e^{x}] equals:

• e^x – (e^x + x e^x) = -x e^x
• e^x – e^x = 0
• -e^x
• x e^x

Correct Answer: -x e^x

Q39. For y = e^{ax} cos bx, dy/dx equals:

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

Correct Answer: a e^{ax} cos bx – b e^{ax} sin bx

Q40. d/dx [\u221a(e^{x})] when \u221a denotes square root, equals:

• e^x / (2 \u221a{e^x})
• \u221a{e^x}
• 1/(2 \u221a x)
• e^{x/2}

Correct Answer: e^x / (2 √{e^x})

Q41. If y = e^{f(x)} and f”(x) exists, which statement is true about y”?

• y” = (f”(x) + (f'(x))^2) e^{f(x)}
• y” = f”(x) e^{f”(x)}
• y” = f'(x) e^{f'(x)}
• y” = e^{f(x)} only

Correct Answer: y” = (f”(x) + (f'(x))^2) e^{f(x)}

Q42. Which is the derivative of e^{x} * e^{2x}?

• 3 e^{3x}
• e^{3x}
• 2 e^{3x}
• e^{x} + e^{2x}

Correct Answer: 3 e^{3x}

Q43. Use logarithmic differentiation: d/dx [x^x e^{x}] equals:

• x^x e^x (ln x + 1 + 1/x)
• x^{x-1} e^x
• x^x e^{x-1}
• x^x e^x ln x

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

Q44. For function y=e^{x}/(1+x), which rule is primarily used to differentiate?

• Quotient rule with chain/product components
• Only product rule
• Only chain rule
• Implicit differentiation

Correct Answer: Quotient rule with chain/product components

Q45. If C(t)=C_0 e^{-kt} and dosing changes C_0, d/dC_0 [C(t)] is:

• e^{-kt}
• -k e^{-kt}
• C_0 e^{-kt}
• 0

Correct Answer: e^{-kt}

Q46. Which derivative identity holds: d/dx [e^{x} g(x)]?

• e^{x} g'(x) + e^{x} g(x)
• g'(x) e^{x} only
• e^{x} g(x)’ only
• e^{x} g(x)”

Correct Answer: e^{x} g'(x) + e^{x} g(x)

Q47. For u(x)=e^{x}, v(x)=\u221ax, d/dx [u/v] at x=1 uses quotient rule. Which is correct?

• (e^1 \u221a1 – e^1(1/(2\u221a1))) / 1
• e / √1
• 0
• e \u221a1

Correct Answer: (e^1 √1 – e^1(1/(2√1))) / 1

Q48. Which is true about the derivative of e^{x} evaluated symbolically?

• d/dx e^{x} = e^{x} for all real x
• d/dx e^{x} = x e^{x} only at x=1
• d/dx e^{x} = 0
• d/dx e^{x} depends on base change

Correct Answer: d/dx e^{x} = e^{x} for all real x

Q49. If y = e^{x} (1 + x)^{-1}, which asymptotic technique helps for derivative at large x?

• Dominant term approximation using derivatives of e^{x}
• Taylor expansion about 0 only
• Ignore e^{x} terms
• Use only logarithmic differentiation

Correct Answer: Dominant term approximation using derivatives of e^{x}

Q50. For drug concentration model C(t)=A e^{bt} + D, dC/dt equals:

• A b e^{bt}
• A e^{bt} + D
• D b e^{bt}
• 0

Correct Answer: A b e^{bt}

Download