Derivative of trigonometric functions (first principles) MCQs With Answer

Introduction: Understanding the derivative of trigonometric functions from first principles is essential for B.Pharm students who apply calculus in pharmacokinetics and drug modeling. This concise guide emphasizes the limit definition, core limits like lim_h→0 (sin h)/h = 1, and step-by-step derivations of derivatives for sin x, cos x, tan x and other trig functions. These trigonometric derivative rules underpin rate-of-change calculations in concentration–time profiles and signal analysis. Practical MCQs help reinforce techniques such as algebraic manipulation, trigonometric identities, and small-angle approximations. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the derivative of sin x using first principles?

• sin x
• cos x
• -sin x
• -cos x

Correct Answer: cos x

Q2. What is the derivative of cos x derived from first principles?

• sin x
• -sin x
• cos x
• -cos x

Correct Answer: -sin x

Q3. Which fundamental limit is essential when deriving the derivative of sin x from first principles?

• lim_h→0 (cos h – 1)/h = 1
• lim_h→0 (sin h)/h = 1
• lim_h→0 (1 – cos h)/h = 1
• lim_h→0 (sin h)/h = 0

Correct Answer: lim_h→0 (sin h)/h = 1

Q4. Using first principles, which additional limit is used to derive the derivative of cos x?

• lim_h→0 (1 – cos h)/h = 0
• lim_h→0 (cos h – 1)/h^2 = 1/2
• lim_h→0 (sin h)/h = 0
• lim_h→0 (tan h)/h = 0

Correct Answer: lim_h→0 (1 – cos h)/h = 0

Q5. What is lim_h→0 (1 – cos h)/h^2?

• 0
• 1
• 1/2
• -1/2

Correct Answer: 1/2

Q6. What is the derivative of tan x obtained using first principles and trig identities?

• sec x
• sec^2 x
• csc^2 x
• tan x sec x

Correct Answer: sec^2 x

Q7. Which identity is commonly used to simplify cos(x+h)-cos x in the first principles derivation?

• cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
• cos A – cos B = -2 sin((A+B)/2) sin((A-B)/2)
• sin A – sin B = 2 cos((A+B)/2) sin((A-B)/2)
• tan A – tan B = sin(A-B)/(cos A cos B)

Correct Answer: cos A – cos B = -2 sin((A+B)/2) sin((A-B)/2)

Q8. Using first principles, derivative of sin(ax) is:

• a cos x
• a cos(ax)
• cos(ax)
• a sin(ax)

Correct Answer: a cos(ax)

Q9. Using first principles, derivative of cos(ax) equals:

• a sin(ax)
• -a sin(ax)
• -sin(ax)
• a cos(ax)

Correct Answer: -a sin(ax)

Q10. From first principles, the derivative of sec x is:

• sec x tan x
• sec^2 x
• tan x
• -sec x tan x

Correct Answer: sec x tan x

Q11. The derivative of csc x (from first principles or using known derivatives) is:

• csc x cot x
• -csc x cot x
• -csc^2 x
• csc^2 x

Correct Answer: -csc x cot x

Q12. The derivative of cot x is:

• -csc^2 x
• csc^2 x
• -sec^2 x
• sec^2 x

Correct Answer: -csc^2 x

Q13. Using first principles, what is the derivative of f(x)=sin^2 x ?

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

Correct Answer: 2 sin x cos x

Q14. Which expression equals the derivative of sin x · cos x?

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

Correct Answer: cos^2 x – sin^2 x

Q15. The small-angle approximation used in first principles is:

• sin x ≈ 1
• sin x ≈ x
• cos x ≈ x
• tan x ≈ 1

Correct Answer: sin x ≈ x

Q16. Using first principles, what is f'(π/2) if f(x)=sin x?

• 1
• 0
• -1
• Undefined

Correct Answer: 0

Q17. Using first principles, f'(0) for f(x)=cos x equals:

• 1
• 0
• -1
• Undefined

Correct Answer: 0

Q18. The derivative of tan x at x = π/4 is:

• 1/2
• 1
• 2
• √2

Correct Answer: 2

Q19. Differentiate f(x)=3 sin x from first principles.

• 3 sin x
• 3 cos x
• -3 sin x
• -3 cos x

Correct Answer: 3 cos x

Q20. Using first principles, derivative of sin(x+y) with respect to x is:

• sin y
• cos(x+y)
• cos x + cos y
• sin x cos y

Correct Answer: cos(x+y)

Q21. If f(x)=sin x / x with f(0)=1, what is f'(0) using the limit definition?

• 1
• 0
• -1
• Undefined

Correct Answer: 0

Q22. Which derivative identity is useful to get derivative of tan x from sin and cos derivatives?

• tan x = sin x · cos x
• tan x = sin x / cos x
• tan x = cos x / sin x
• tan x = sec x · cos x

Correct Answer: tan x = sin x / cos x

Q23. The derivative of sin(2x) from first principles is:

• 2 cos x
• 2 cos(2x)
• cos(2x)
• sin(2x)

Correct Answer: 2 cos(2x)

Q24. The derivative of sec(3x) equals:

• 3 sec(3x) tan(3x)
• sec(3x) tan(3x)
• 3 sec^2(3x)
• sec^2(3x)

Correct Answer: 3 sec(3x) tan(3x)

Q25. Which limit value equals lim_h→0 (cos h – 1)/h ?

• 0
• 1
• -1/2
• Undefined

Correct Answer: 0

Q26. What is lim_h→0 (cos h – 1)/h^2 ?

• 1/2
• -1/2
• 0
• 1

Correct Answer: -1/2

Q27. The derivative of arcsin x (inverse sine) is:

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

Correct Answer: 1/√(1-x^2)

Q28. The derivative of arctan x is:

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

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

Q29. The derivative of arccos x is:

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

Correct Answer: -1/√(1-x^2)

Q30. Is sin x differentiable for all real x?

• Yes, differentiable everywhere
• No, differentiable only where x is integer multiple of π
• No, differentiable only in (−π/2, π/2)
• Only differentiable at x=0

Correct Answer: Yes, differentiable everywhere

Q31. Where is csc x not differentiable?

• At x = π/2 + kπ
• At x = kπ (integer multiples of π)
• Everywhere
• Nowhere

Correct Answer: At x = kπ (integer multiples of π)

Q32. Using first principles, derivative of sin^3 x is:

• 3 sin^2 x cos x
• sin^2 x
• 3 sin x cos x
• cos^3 x

Correct Answer: 3 sin^2 x cos x

Q33. Using first principles and trig identities, derivative of sin x + cos x equals:

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

Correct Answer: cos x – sin x

Q34. The derivative of tan(2x) is:

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

Correct Answer: 2 sec^2(2x)

Q35. Which expression is a correct step when deriving derivative of sin x using first principles?

• sin(x+h)-sin x = 2 sin(x+h)/2 cos(x-h)/2
• sin(x+h)-sin x = 2 cos((2x+h)/2) sin(h/2)
• sin(x+h)-sin x = 2 cos((2x+h)/2) cos(h/2)
• sin(x+h)-sin x = 2 cos((2x+h)/2) sin(h/2)

Correct Answer: sin(x+h)-sin x = 2 cos((2x+h)/2) sin(h/2)

Q36. Evaluate derivative f'(π) for f(x)=sin x using first principles.

• 1
• 0
• -1
• Undefined

Correct Answer: -1

Q37. If f(x)=sin x / cos x, from first principles the derivative is:

• sec x
• sec^2 x
• csc^2 x
• tan x sec x

Correct Answer: sec^2 x

Q38. Which manipulation helps reduce (sin(x+h)-sin x)/h to cos x as h→0?

• Use sin(A)-sin(B) = 2 cos((A+B)/2) sin((A-B)/2)
• Replace sin by tan directly
• Split into cos terms only
• Use double-angle for cosine

Correct Answer: Use sin(A)-sin(B) = 2 cos((A+B)/2) sin((A-B)/2)

Q39. The derivative of f(x)=a sin x + b cos x is:

• a cos x + b sin x
• a cos x – b sin x
• -a sin x + b cos x
• -a cos x – b sin x

Correct Answer: a cos x – b sin x

Q40. True or False: The derivative of sec x exists at x = π/2.

• True
• False
• Only if sec x is redefined
• Depends on one-sided limits

Correct Answer: False

Q41. Differentiate f(x)=sin(3x+π/4) using first principles approach to scaling and shift.

• 3 cos(3x+π/4)
• cos(3x+π/4)
• -3 sin(3x+π/4)
• 3 sin(3x+π/4)

Correct Answer: 3 cos(3x+π/4)

Q42. Using first principles, the derivative of g(x)=cos(5x) is:

• 5 sin(5x)
• -5 sin(5x)
• -sin(5x)
• 5 cos(5x)

Correct Answer: -5 sin(5x)

Q43. The derivative of f(x)=tan x + sec x is:

• sec^2 x + sec x tan x
• sec x + tan x
• sec^2 x – sec x tan x
• tan^2 x + sec^2 x

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

Q44. Which of these is NOT a step in deriving derivative of cos x from first principles?

• Use cos(x+h)-cos x identity
• Divide by h and take limit h→0
• Apply limit lim (sin h)/h = 1
• Replace cos x by sin x directly without identity

Correct Answer: Replace cos x by sin x directly without identity

Q45. The derivative of y = sin x / cos x at x=0 is:

• 0
• 1
• Undefined
• 2

Correct Answer: 1

Q46. Using first principles, derivative of h(x)=cos x · cos x (i.e., cos^2 x) is:

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

Correct Answer: -2 cos x sin x

Q47. When deriving trig derivatives from limits, which two basic limits are typically applied?

• lim (sin h)/h = 1 and lim (tan h)/h = 0
• lim (sin h)/h = 1 and lim (1 – cos h)/h = 0
• lim (cos h – 1)/h = 1 and lim (sin h)/h = 0
• lim (1 – cos h)/h^2 = 2 and lim (sin h)/h = 1

Correct Answer: lim (sin h)/h = 1 and lim (1 – cos h)/h = 0

Q48. The derivative of y = sin x at x = π is:

• 0
• 1
• -1
• Undefined

Correct Answer: -1

Q49. Which statement is false regarding trigonometric derivatives?

• Derivative of sin x is cos x
• Derivative of cos x is -sin x
• Derivative of sec x is sec x tan x
• Derivative of tan x is csc^2 x

Correct Answer: Derivative of tan x is csc^2 x

Q50. Using first principles and known limits, the derivative at 0 of f(x) = sin(2x)/x after defining f(0)=2 is:

• 0
• 1
• 2
• Undefined

Correct Answer: 0

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