# Continuity and Differentiability Notes | SwaVid Continuity and Differentiability, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/continuity-and-differentiability Source: https://www.swavid.com/cbse/class-12/maths/chapter/continuity-and-differentiability # Continuity and Differentiability ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 12 Mathematics on SwaVid ### Continuity of a function ### Differentiability and continuity ### Chain rule ### Derivatives of inverse trigonometric functions ### Logarithmic and exponential differentiation ### Parametric differentiation ### Second order derivative ### Next chapter ### Related chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 12 › Mathematics › Continuity and Differentiability Notes written by SwaVid from the NCERT textbook. Class 12 Mathematics, Chapter 5. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter explains the concepts of continuity of functions, differentiability at a point, and rules for differentiating algebraic, inverse trigonometric, exponential, and logarithmic functions. Students learn how to apply the chain rule, implicit differentiation, logarithmic differentiation, and find second order derivatives. Source: NCERT Mathematics Part I (Reprint 2026-27) , Chapter 5, Continuity and Differentiability A real function f is continuous at a point c in its domain if the limit of f(x) as x approaches c equals f(c). A function is continuous in an interval if it is continuous at every point in that interval. A function f is differentiable at c if the limit of (f(c + h) - f(c)) / h as h approaches 0 exists finitely. Every differentiable function is continuous, but the converse is not necessarily true. To differentiate a composite function f(g(x)), differentiate f with respect to g(x) and multiply the result by the derivative of g(x) with respect to x. In Leibniz notation, dy/dx = (dy/du) × (du/dx). The chapter uses implicit differentiation and trigonometric identities to establish derivatives of inverse trigonometric functions like sin⁻¹ x, cos⁻¹ x, and tan⁻¹ x within their valid domains. The derivative of eˣ is eˣ and the derivative of ln x is 1/x for x > 0. Taking natural logarithms on both sides simplifies functions of the form u(x) raised to the power v(x) before differentiating. When x and y are given in terms of a third parameter t, dy/dx is computed as (dy/dt) / (dx/dt), provided dx/dt is not zero. The second order derivative d²y/dx² is obtained by differentiating dy/dx once more with respect to x. For parametric forms, d²y/dx² = [d/dt (dy/dx)] × (dt/dx). Try each question first. Then tap it to see the answer. Left hand limit and value at x = 2 is f(2) = k × (2)² = 4k Right hand limit as x → 2⁺ is lim (x → 2⁺) 3 = 3 For continuity, 4k = 3 k = 3/4 k = 3/4 Compute dx/dθ = -a sin θ Compute dy/dθ = a cos θ dy/dx = (dy/dθ) / (dx/dθ) = (a cos θ) / (-a sin θ) dy/dx = -cot θ -cot θ Differentiate once: dy/dx = 3x² + sec² x Differentiate again: d²y/dx² = d/dx (3x²) + d/dx (sec² x) d/dx (3x²) = 6x d/dx (sec² x) = 2 sec x × (sec x tan x) = 2 sec² x tan x Combine the terms: d²y/dx² = 6x + 2 sec² x tan x 6x + 2 sec² x tan x Application of Derivatives . Integrals . Inverse Trigonometric Functions . The 13 chapters of Class 12 Mathematics . CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Application of Derivatives. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 5 NCERT book - 80 marks, theory paper - 3 quick checks - Applying the power rule to functions of the form u(x) to the power v(x) . Do not treat the exponent as a constant. Take natural logarithms on both sides first, or rewrite the function as e^[v(x) ln u(x)] before taking the derivative. - Calculating d²y/dx² in parametric form as (d²y/dt²) / (d²x/dt²) . Differentiate dy/dx with respect to t, then multiply by dt/dx. The correct expression is [d/dt (dy/dx)] / (dx/dt). - Assuming continuity automatically implies differentiability . A function can be continuous at a point without being differentiable there, such as f(x) = |x| at x = 0. Verify the left-hand and right-hand derivatives separately. - Forgetting the inner derivative when using the chain rule . Always multiply by the derivative of the interior function, such as d/dx [sin(x²)] = cos(x²) × 2x, not just cos(x²). ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 12](https://www.swavid.com/cbse/class-12) - [Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 13 chapters](https://www.swavid.com/cbse/class-12/maths/chapters) - [NCERT Mathematics Part I (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/application-of-derivatives) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/integrals) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/inverse-trigonometric-functions) - [Class 12 Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup)