Class 11 Maths Chapter 12 Limits and Derivatives – Extra Questions with Answers

Finding a derivative from first principles means evaluating a limit as a small change in x shrinks toward zero. This set of Class 11 Maths Chapter 12 questions works through direct limit evaluation and basic derivative rules for polynomials and trigonometric functions.

Last Updated: September 23, 2026

Very Short Answer Questions (1 mark)

Q1. Find lim_x→2 (x²+3).
Ans: 2²+3 = 7.

Q2. Find the derivative of x⁵.
Ans: 7x⁶.

Q3. Find the derivative of a constant, e.g. f(x)=5.
Ans: 0.

Q4. What is d/dx(sin x)?
Ans: cos x.

Q5. Evaluate lim_x→0 (1−cos x)/x.
Ans: 0.

Short Answer Questions (2–3 marks)

Q6. Evaluate lim_x→3 (x²−9)/(x−3), noting the expression is indeterminate at x=3 initially.
Ans: Factor: (x²−9)/(x−3) = (x+3)(x−3)/(x−3) = x+3 (for x≠3). So lim_x→3 = 3+3 = 6.

Q7. Find the derivative of f(x) = 3x² + 5x using the standard derivative rules.
Ans: f'(x) = 3(2x) + 5(1) = 6x + 5.

Q8. Using the first-principles (limit) definition, find the derivative of f(x) = x².
Ans: f'(x) = lim_h→0 [(x+h)²−x²]/h = lim_h→0 [x²+2xh+h²−x²]/h = lim_h→0 [2xh+h²]/h = lim_h→0 (2x+h) = 2x.

Higher-Order Thinking / Application Questions

Q9. Explain, in conceptual terms, why the derivative is defined as a limit rather than a simple ratio like [f(x+h)−f(x)]/h evaluated at some fixed small h.
Ans: Using a fixed small h would only give an approximate average rate of change over that small interval, not the true instantaneous rate of change at a single point. By taking the limit as h approaches 0 (rather than plugging in a specific small value), we capture the exact, precise rate of change at that exact point, free from the choice of any particular interval size.

Q10. A car’s position is given by s(t) = t² + 2t (in metres, t in seconds). Find its velocity function using derivatives, and determine the velocity at t=3 seconds. Explain the physical meaning of the derivative in this context.
Ans: Velocity v(t) = s'(t) = 2t + 2. At t=3: v(3) = 2(3)+2 = 8 m/s. Physically, the derivative of position with respect to time gives the instantaneous velocity — the exact speed and direction of motion at that precise moment, rather than an average speed over some time interval.

Frequently Asked Questions

How would you evaluate the limit of (x squared minus 4) over (x minus 2) as x approaches 2?
Factor the numerator as (x-2)(x+2), cancel the common term, and the limit becomes x+2 evaluated at 2, which is 4.

How would you find the derivative of f(x) = x cubed using first principles?
Using the definition f prime of x = limit as h approaches 0 of ((x+h) cubed minus x cubed) over h, expanding gives 3x squared after simplification.

Written by Satish

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