Derivative of loge x MCQs With Answer

Introduction: The derivative of loge x (ln x) is a fundamental concept for B.Pharm students studying pharmacokinetics, drug stability, and concentration–time modeling. Understanding d/dx ln x = 1/x (for x > 0) and the more general d/dx ln|x| = 1/x (x ≠ 0) helps analyze relative rates, half-life calculations, pH-related transformations, and exponential decay. This topic links calculus rules — chain rule, product rule, and implicit differentiation — to practical pharmaceutical problems. These MCQs with answers reinforce both theory and application, improving problem-solving for dosing, kinetics and lab calculations. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the derivative of loge x with respect to x for x > 0?

• 1/x
• ln x
• x ln x
• x

Correct Answer: 1/x

Q2. The derivative of ln|x| is valid for which values of x?

• All real x except x = 0
• Only x > 0
• Only x < 0
• All real x including x = 0

Correct Answer: All real x except x = 0

Q3. Using chain rule, what is d/dx [ln(u(x))]?

• u(x) / u'(x)
• u'(x) / u(x)
• ln u'(x)
• u'(x) * ln u(x)

Correct Answer: u'(x) / u(x)

Q4. What is d/dx [ln(3x^2 + 1)]?

• 6x / (3x^2 + 1)
• 3x^2 / (3x^2 + 1)
• 6x ln(3x^2 + 1)
• 1 / (3x^2 + 1)

Correct Answer: 6x / (3x^2 + 1)

Q5. What is d/dx [ln(ax)] where a is a positive constant?

• 1 / (ax)
• a / x
• 1 / x
• ln a / x

Correct Answer: 1 / x

Q6. What is the derivative of ln(x^2)?

• 2 / x
• 1 / x^2
• 2x / ln x
• ln x / x

Correct Answer: 2 / x

Q7. For y = ln(f(x)) where f(x) = e^{2x}, what is dy/dx?

• 2
• e^{2x}
• 2e^{2x}
• 1/2

Correct Answer: 2

Q8. d/dx [ln(x) + ln(y)] with x a function of t and y constant: which term remains?

• d/dx ln(y)
• 1/x
• 0
• ln y

Correct Answer: 1/x

Q9. What is d/dx [ln(1/x)]?

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

Correct Answer: -1 / x

Q10. Which expression equals d/dx [ln|x|] at x = -2?

• -1/2
• 1/2
• -2
• Undefined

Correct Answer: -1/2

Q11. The second derivative of ln x is:

• -1 / x^2
• 1 / x^2
• 0
• 1 / x

Correct Answer: -1 / x^2

Q12. d/dx [ln(sin x)] for 0 < x < π is:

• cos x / sin x
• sin x / cos x
• 1 / sin x
• ln(cos x)

Correct Answer: cos x / sin x

Q13. If C(t) = C0 e^{-kt}, what is d/dt [ln C(t)]?

• -k
• k
• -k C0
• e^{-kt}

Correct Answer: -k

Q14. Which rule is most directly used to differentiate ln(g(x))?

• Chain rule
• Product rule
• Integration by parts
• Quotient rule

Correct Answer: Chain rule

Q15. d/dx [ln(x) * x] requires which combination of rules?

• Product rule only
• Chain rule only
• Product rule and derivative of ln (1/x)
• Quotient rule and chain rule

Correct Answer: Product rule and derivative of ln (1/x)

Q16. What is d/dx [ln(x) / x]?

• (1 – ln x) / x^2
• (ln x – 1) / x^2
• 1 / x^2
• ln x / x

Correct Answer: (1 – ln x) / x^2

Q17. For y = ln(u) where u = x^n, dy/dx equals:

• n / x
• x^{n-1} / n
• ln x / n

Correct Answer: n / x

Q18. d/dx [log_a x] equals:

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

Correct Answer: 1 / (x ln a)

Q19. If y = ln(x) and x = e^t, dy/dt =

• 1
• e^t
• t
• 1 / e^t

Correct Answer: 1

Q20. Differentiate y = ln(x+√(x^2+1)). dy/dx =

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

Correct Answer: 1 / √(x^2+1)

Q21. For drug concentration C, instantaneous relative rate r = (1/C) dC/dt equals:

• d/dt [ln C]
• ln(dC/dt)
• C * dC/dt
• d/dt [1/C]

Correct Answer: d/dt [ln C]

Q22. Which is the derivative of ln(uv) where u and v are functions of x?

• u’ / u + v’ / v
• (uv)’ / uv
• ln u’ + ln v’
• u’v + uv’

Correct Answer: u’ / u + v’ / v

Q23. d/dx [ln(x^2 + 1)^{3}] simplifies to:

• 6x / (x^2 + 1)
• 3 / (x^2 + 1)
• 3 ln(x^2 + 1) / x
• 6x ln(x^2 + 1)

Correct Answer: 6x / (x^2 + 1)

Q24. Using logarithmic differentiation, d/dx [x^x] equals:

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

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

Q25. For f(x) = ln|sin x|, where is f'(x) undefined?

• At x = nπ where n is integer
• At x = π/2
• At x = π/4
• Nowhere; always defined

Correct Answer: At x = nπ where n is integer

Q26. What is d/dx [ln( (x+1)/(x-1) )]?

• 2 / (x^2 – 1)
• 1 / (x+1) – 1 / (x-1)
• (x-1 – x-1) / (x^2-1)
• ln((x+1)/(x-1))’ = 0

Correct Answer: 2 / (x^2 – 1)

Q27. Which statement is true about ln x at x = 0?

• ln x is not defined at x = 0
• ln x has derivative 1 at x = 0
• ln x has a finite value at x = 0
• ln x is defined and differentiable at x = 0

Correct Answer: ln x is not defined at x = 0

Q28. If y = ln(g(x)) and g(a) = 1, what is y'(a)?

• g'(a)
• 0
• g'(a) / 1 = g'(a)
• 1

Correct Answer: g'(a) / 1 = g'(a)

Q29. Differentiate y = ln( e^{x} + e^{-x} ). y’ =

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

Correct Answer: (e^{x} – e^{-x}) / (e^{x} + e^{-x})

Q30. For concentration C(t) = A / (1 + Bt), what is d/dt [ln C]?

• -B / (1 + Bt)
• B / (1 + Bt)
• -A / (1 + Bt)^2
• 0

Correct Answer: -B / (1 + Bt)

Q31. The derivative of ln(x) at x = 1 is:

• 1
• 0
• -1
• Undefined

Correct Answer: 1

Q32. d/dx [ln(x) * ln(2x)] requires which simplification step?

• Use product rule and derivative 1/x and 1/x for ln terms
• Treat ln(2x) as constant
• Differentiate ln(x) only
• Use quotient rule only

Correct Answer: Use product rule and derivative 1/x and 1/x for ln terms

Q33. If f(x) = ln(g(x))/h(x), which rule is used to find f'(x)?

• Quotient rule combined with chain rule
• Simple chain rule only
• Product rule only
• Integration

Correct Answer: Quotient rule combined with chain rule

Q34. d/dx [ln(1 + e^{x})] equals:

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

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

Q35. For f(x) = ln( (x^3) ), f'(x) simplifies to:

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

Correct Answer: 3 / x

Q36. Which identity helps convert differentiation of ln(a x^n) quickly?

• ln(a x^n) = ln a + n ln x
• ln(a x^n) = a + n x
• ln(a x^n) = a ln x^n
• ln(a x^n) = ln a * ln x^n

Correct Answer: ln(a x^n) = ln a + n ln x

Q37. If y = ln(u) and u = v^k, dy/dx =

• k v’ / v
• v’ / v^k
• k ln v
• v^k / v’

Correct Answer: k v’ / v

Q38. What is derivative of ln(1 – x) with respect to x?

• -1 / (1 – x)
• 1 / (1 – x)
• – (1 – x)
• ln(1 – x)’ = 0

Correct Answer: -1 / (1 – x)

Q39. For a dissolution rate proportional to concentration C, d/dt ln C = constant. This constant represents:

• The proportional rate constant
• Absolute change in concentration
• Half-life
• Initial concentration

Correct Answer: The proportional rate constant

Q40. d/dx [ln(x) + ln(2x)] equals:

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

Correct Answer: 1/x + 1/x

Q41. Evaluate derivative at x = 2: d/dx [ln(x^2 + x)] at x=2 equals:

• (2x + 1)/(x^2 + x) at x=2 = 5/6
• 1/2
• 2/(x^2 + x) at x=2 = 2/6
• ln(6)/2

Correct Answer: (2x + 1)/(x^2 + x) at x=2 = 5/6

Q42. d/dx [ln|x|] equals 1/x. This implies the slope sign changes at:

• Across x = 0, slope goes from negative to positive depending on sign of x
• x = 1 only
• Never; slope is constant
• x = -1 only

Correct Answer: Across x = 0, slope goes from negative to positive depending on sign of x

Q43. If pH = -log10[H+], express dpH/d[H+] using natural log relation. dpH/d[H+] =

• -1 / ([H+] ln 10)
• -ln 10 / [H+]
• 1 / ([H+] ln 10)
• -1 / [H+]

Correct Answer: -1 / ([H+] ln 10)

Q44. Differentiate y = ln(√x). y’ equals:

• 1 / (2x)
• 1 / x
• 1 / (√x)
• ln √x / x

Correct Answer: 1 / (2x)

Q45. For y = ln( (x^2 + 1)^{1/2} ), y’ simplifies to:

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

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

Q46. Using logarithmic differentiation for f(x) = (x+1)^{x}, f'(x) includes which main term?

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

Correct Answer: (x+1)^{x} (ln(x+1) + x/(x+1))

Q47. The derivative of ln x approaches what value as x → ∞?

• 0
• 1
• ∞
• -∞

Correct Answer: 0

Q48. Differentiate y = ln( (x^2 – 1) ) for x > 1. y’ equals:

• 2x / (x^2 – 1)
• 1 / (x^2 – 1)
• ln(x^2 – 1)’ = 0
• 2 / (x – 1)

Correct Answer: 2x / (x^2 – 1)

Q49. If concentration follows C = C0 t^{-n}, ln C = ln C0 – n ln t. d/dt ln C equals:

• -n / t
• n / t
• -n t^{n-1}
• 0

Correct Answer: -n / t

Q50. Which practical interpretation is correct: d/dt [ln(concentration)] gives

• Instantaneous fractional (relative) rate of change of concentration
• Absolute concentration change per unit time
• Half-life directly
• Initial concentration

Correct Answer: Instantaneous fractional (relative) rate of change of concentration

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