Derivative of ax MCQs With Answer

Understanding the derivative of ax and related exponential forms is essential for B. Pharm students studying pharmacokinetics and drug-reaction rates. This concise guide focuses on differentiation rules for linear expressions (ax), power functions, and exponential functions like a^x and e^x, plus chain, product, and logarithmic differentiation methods. Emphasis is placed on practical applications—rate of change of drug concentration, half-life calculations, and slope interpretation—so students can apply calculus directly to dosage and concentration models. Keywords included: derivative of ax, derivative of a^x, differentiation rules, pharmacokinetics, chain rule, logarithmic differentiation. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the derivative with respect to x of the linear function ax (where a is a constant)?

• a + x
• ax
• a
• 0

Correct Answer: a

Q2. If f(x) = a^x with constant a > 0, what is f'(x)?

• a^x
• a^x ln a
• x a^{x-1}
• ln x · a^x

Correct Answer: a^x ln a

Q3. For f(x) = e^x, the derivative f'(x) equals:

• e^x ln e
• x e^{x-1}
• e^x
• ln x

Correct Answer: e^x

Q4. What is the derivative of f(x) = ax^n (a constant, n a positive integer)?

• an x^{n-1}
• a x^{n+1}
• n x^{a-1}
• a n x^{n}

Correct Answer: an x^{n-1}

Q5. If C(t) = C0 e^{-kt} represents drug concentration, what is dC/dt?

• -k C0 e^{-kt}
• k C0 e^{-kt}
• C0 e^{kt}
• -C0 e^{-kt} / k

Correct Answer: -k C0 e^{-kt}

Q6. The derivative of f(x) = ln x is:

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

Correct Answer: 1 / x

Q7. Using chain rule, derivative of f(x) = a^{g(x)} is:

• a^{g(x)} g'(x) / a
• a^{g(x)} ln a · g'(x)
• g'(x) a^{g(x)-1}
• ln(g(x)) a^{g(x)}

Correct Answer: a^{g(x)} ln a · g'(x)

Q8. If y = x^3 · e^{2x}, what rule best finds dy/dx?

• Quotient rule
• Product rule combined with chain rule
• Power rule only
• Logarithmic differentiation only

Correct Answer: Product rule combined with chain rule

Q9. The derivative of f(x) = a (constant) is:

• a x
• 0
• 1
• ln a

Correct Answer: 0

Q10. If f(x) = 1^x, what is f'(x)?

• 1
• 0
• ln 1
• 1^x ln 1

Correct Answer: 0

Q11. For f(x) = a^x and a = e, which identity simplifies the derivative?

• ln e = 1
• e^x = ln x
• a^x = x^a
• ln a = 0

Correct Answer: ln e = 1

Q12. Differentiate y = (a^x)^2. Which is correct?

• 2 a^{2x} ln a
• a^{2x} ln a
• 2x a^{2x-1}
• 2 a^x ln a

Correct Answer: 2 a^{2x} ln a

Q13. If f(x) = x a^x (a constant), f'(x) equals:

• a^x + x a^x ln a
• a^x (1 + ln a)
• x a^{x-1} + a^x
• x a^x

Correct Answer: a^x + x a^x ln a

Q14. The derivative of f(x) = a^{2x+1} is:

• a^{2x+1} · 2 ln a
• 2 a^{2x} ln a
• a^{2x+1} ln(2x+1)
• 2x a^{2x+1}

Correct Answer: a^{2x+1} · 2 ln a

Q15. Which differentiation technique is most useful for y = x^x?

• Power rule directly
• Logarithmic differentiation
• Quotient rule
• Integration by parts

Correct Answer: Logarithmic differentiation

Q16. If y = a^{ln x} (a constant), y simplifies to which function before differentiating?

• x^{ln a}
• a^x ln x
• e^{ln a · ln x} = x^{ln a}
• ln(a ln x)

Correct Answer: e^{ln a · ln x} = x^{ln a}

Q17. The second derivative of f(x) = e^{kx} is:

• k^2 e^{kx}
• k e^{kx}
• e^{k x} / k
• k^2 x e^{kx}

Correct Answer: k^2 e^{kx}

Q18. For pharmacokinetics, clearance rate dC/dt proportional to C gives which differential form?

• dC/dt = k
• dC/dt = -k C
• dC/dt = C / k
• dC/dt = -k

Correct Answer: dC/dt = -k C

Q19. Derivative with respect to x of f(x) = a^{x^2} is:

• a^{x^2} · 2x ln a
• 2x a^{x^2}
• x^2 a^{x^2-1}
• a^{x^2} ln(x^2)

Correct Answer: a^{x^2} · 2x ln a

Q20. If f(x) = ln(a^x), simplify f'(x):

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

Correct Answer: ln a

Q21. The derivative of f(x) = ax + b (a, b constants) is:

• a x + b
• a
• b
• 0

Correct Answer: a

Q22. If f(x) = a^{u(x)} and a = 10, the derivative includes which factor?

• ln 10
• log_{10} e
• 1 / ln 10
• ln u(x)

Correct Answer: ln 10

Q23. For y = x^2 · a^{3x}, what is an element appearing in dy/dx?

• x^2 · a^{3x} ln a · 3
• x^2 · 3 a^{3x-1}
• 2x a^{3x} / ln a
• ln x · a^{3x}

Correct Answer: x^2 · a^{3x} ln a · 3

Q24. Differentiate y = a^{x} / x. Which rule is primarily used?

• Product rule
• Quotient rule
• Power rule only
• Trapezoidal rule

Correct Answer: Quotient rule

Q25. If y = x ln a (a constant), dy/dx equals:

• ln a
• x / a
• a x
• 0

Correct Answer: ln a

Q26. The derivative of f(x) = a^{x} · ln a equals:

• a^{x}
• a^{x} (ln a)^2
• ln a
• x a^{x-1} ln a

Correct Answer: a^{x} (ln a)^2

Q27. For y = (ax)^n, which method helps simplify differentiation?

• Expand then differentiate only for all n
• Use constant multiple and power rule: derivative = n (a x)^{n-1} · a
• Use derivative of a^x
• Use partial fractions

Correct Answer: Use constant multiple and power rule: derivative = n (a x)^{n-1} · a

Q28. If y = a^{x} and a < 1 (e.g., a = 0.5), f'(x) is:

• Negative because a < 1
• a^{x} ln a which is negative
• a^{x} / ln a
• Zero

Correct Answer: a^{x} ln a which is negative

Q29. The derivative of f(x) = log_a x equals:

• 1 / (x ln a)
• ln a / x
• 1 / x
• ln x / a

Correct Answer: 1 / (x ln a)

Q30. Using logarithmic differentiation, derivative of y = (sin x)^{a} (a constant) gives:

• a (sin x)^{a-1} cos x
• (sin x)^{a} ln sin x
• a (sin x)^{a} cos x
• ln a · (sin x)^a

Correct Answer: a (sin x)^{a-1} cos x

Q31. If C(t) = C0 a^{kt} models concentration growth, dC/dt is:

• C0 a^{kt} k ln a
• C0 k a^{kt-1}
• k C0
• a^{kt} / k

Correct Answer: C0 a^{kt} k ln a

Q32. The derivative of y = x · ln a (a constant) gives a slope equal to:

• Dependent on x
• ln a (constant)
• a
• 1 / ln a

Correct Answer: ln a (constant)

Q33. If y = a^{f(x)} and f(x) = ln x, dy/dx simplifies to:

• a^{ln x} · (1/x) ln a
• a^{ln x} / x
• a^{ln x} ln(ln x)
• 1 / (x ln a)

Correct Answer: a^{ln x} · (1/x) ln a

Q34. For y = e^{g(x)} where g'(x) = 3x^2, dy/dx equals:

• e^{g(x)}
• 3x^2 e^{g(x)}
• g(x) e^{g(x)}
• e^{3x^2}

Correct Answer: 3x^2 e^{g(x)}

Q35. The derivative of f(x) = a^{x} + b^{x} (a,b constants) is:

• a^{x} ln a + b^{x} ln b
• (a+b)^x ln(a+b)
• x a^{x-1} + x b^{x-1}
• ln(a b) (a^{x} + b^{x})

Correct Answer: a^{x} ln a + b^{x} ln b

Q36. If y = x / a^x, which derivative component appears after quotient rule?

• -x a^{x} ln a in numerator
• a^{x} in denominator only
• ln x in numerator
• 1 / (a^x ln a)

Correct Answer: -x a^{x} ln a in numerator

Q37. A derivative test: if f'(x) = 0 for f(x) = ax, what does that imply about a?

• a = 0
• x = 0
• a is variable
• a = 1

Correct Answer: a = 0

Q38. Differentiate y = a^{x} where a = e^k (k constant). Then y’ equals:

• a^{x} k
• a^{x} ln a
• a^{x} k x
• e^{kx} ln k

Correct Answer: a^{x} ln a

Q39. If f(x) = (a^x – 1)/x, what limit-based derivative concept may be used at x→0?

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

Correct Answer: L’Hôpital’s rule

Q40. The derivative of y = a^{m x + c} with constants m,c is:

• a^{m x + c} · m ln a
• m a^{m x + c}
• a^{m x} · c
• a^{m x + c} ln(m x + c)

Correct Answer: a^{m x + c} · m ln a

Q41. For y = e^{ax} / x, which term arises after differentiation?

• -e^{ax} / x^2 + a e^{ax} / x
• e^{ax} / x only
• a x e^{ax}
• -a e^{ax} / x^2

Correct Answer: -e^{ax} / x^2 + a e^{ax} / x

Q42. Differentiate f(x) = a^{x} · b^{x} (a,b constants). The derivative is:

• (ab)^{x} (ln a + ln b)
• a^{x} b^{x} ln(ab)
• a^{x} ln a + b^{x} ln b
• (a+b)^{x} ln(a+b)

Correct Answer: a^{x} b^{x} ln(ab)

Q43. The derivative of f(x) = x^n ln a (a constant) is:

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

Correct Answer: n x^{n-1} ln a

Q44. If y = a^{x^3}, what is dy/dx at x = 0?

• a^0 · 3x^2 ln a evaluated at 0 → 0
• ln a
• 3 ln a
• 1

Correct Answer: a^0 · 3x^2 ln a evaluated at 0 → 0

Q45. For f(x) = ln(a^x + 1), f'(x) contains which factor?

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

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

Q46. If y = a^{x} with a>0, the instantaneous rate of change at x=0 equals:

• a^0 ln a = ln a
• a
• 0
• 1

Correct Answer: a^0 ln a = ln a

Q47. The derivative of f(x) = (a^x)^{b} (b constant) simplifies to:

• b a^{bx} ln a
• a^{bx} ln b
• b a^{x(b-1)}
• a^{x b} / b

Correct Answer: b a^{bx} ln a

Q48. Using differentiation, half-life t1/2 from C(t) = C0 e^{-kt} yields which relation?

• t1/2 = ln 2 / k
• t1/2 = k / ln 2
• t1/2 = ln k / 2
• t1/2 = 2 / k

Correct Answer: t1/2 = ln 2 / k

Q49. For y = a^{x} with a variable a(x), total derivative dy/dx is:

• a'(x) x a^{x-1}
• a^{x} (ln a · x’ + a’/a · x) — not standard
• ∂/∂x a^{x} treating a constant
• Use partial derivatives: dy = a^{x} ln a · dx + x a^{x-1} da

Correct Answer: Use partial derivatives: dy = a^{x} ln a · dx + x a^{x-1} da

Q50. For practical B.Pharm application, which derivative describes instantaneous rate of infusion if amount A(t) = R t?

• dA/dt = R (infusion rate constant)
• dA/dt = t R’
• dA/dt = R t^2
• dA/dt = 0

Correct Answer: dA/dt = R (infusion rate constant)

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