Schooling · Class 12 · Mathematics · Continuity and Differentiability · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-I
Continuity and Differentiability: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Continuity at a point
- Differentiability at a point
- Differentiability implies continuity
- Chain rule
- Implicit differentiation
- Logarithmic differentiation
- Parametric differentiation
- Formulas and facts
- d/dx (xⁿ) = n x^(n−1)
- d/dx (eˣ) = eˣ
- d/dx (aˣ) = aˣ ln a
- d/dx (ln x) = 1/x
- Formula / Rule
- d/dx (tan⁻¹ x) = 1/(1 + x²)
- Worked examples
- Testing continuity
- Implicit differentiation
- Logarithmic differentiation
- Watch out for
- Forgetting to check that f(a) is…
- Assuming continuity guarantees…
- Omitting the inner derivative in the…
- Treating y as a constant in implicit…
- Confusing the Mean Value Theorem…
What this chapter is about
This chapter builds the theoretical foundation for calculus by formalising two connected ideas: continuity (a function has no breaks or jumps) and differentiability (a function has a well-defined slope at each point). In Class 11 you met limits and the idea of a derivative; now you study when derivatives exist and how to compute them for a wide range of functions.
Key ideas
- Continuity at a point: A function f is continuous at x = a when lim x→a f(x) exists, f(a) is defined, and lim x→a f(x) = f(a). Informally, you can draw the curve through a without lifting your pen.
- Differentiability at a point: f is differentiable at x = a when lim h→0 [f(a + h) − f(a)] / h exists and is finite. This limit, if it exists, is the derivative f ′(a).
- Differentiability implies continuity: If f is differentiable at a point, it must be continuous there; the converse is false (a sharp corner can be continuous but not differentiable).
- Chain rule: For a composite y = f(g(x)), the derivative is dy/dx = f ′(g(x)) · g ′(x). In Leibniz form, dy/dx = (dy/du)(du/dx) where u = g(x).
- Implicit differentiation: When y is not isolated (e.g., x² + y² = 25), differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
- Logarithmic differentiation: For products, quotients or variable exponents, take ln of both sides, differentiate, then multiply back by the original function.
- Parametric differentiation: If x = φ(t) and y = ψ(t), then dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
- Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) with f ′(c) = [f(b) − f(a)] / (b − a). Rolle's theorem is the special case when f(a) = f(b), giving f ′(c) = 0.
Formulas and facts to remember
- Formula / Rule: d/dx (xⁿ) = n x^(n−1) · Meaning: Power rule, valid for any real n.
- Formula / Rule: d/dx (eˣ) = eˣ · Meaning: Exponential function is its own derivative.
- Formula / Rule: d/dx (aˣ) = aˣ ln a · Meaning: Base-a exponential, a > 0, a ≠ 1.
- Formula / Rule: d/dx (ln x) = 1/x · Meaning: Natural logarithm, x > 0.
- Formula / Rule: d/dx (sin x) = cos x ; d/dx (cos x) = −sin x · Meaning: Basic trigonometric derivatives.
- Formula / Rule: d/dx (tan x) = sec² x ; d/dx (cot x) = −cosec² x · Meaning:
- Formula / Rule: d/dx (sin⁻¹ x) = 1/√(1 − x²) ; d/dx (cos⁻¹ x) = −1/√(1 − x²) · Meaning: Inverse sine and cosine, |x| < 1.
- Formula / Rule: d/dx (tan⁻¹ x) = 1/(1 + x²) · Meaning: Inverse tangent, all real x.
- Formula / Rule: Product rule: d/dx (uv) = u (dv/dx) + v (du/dx) · Meaning:
- Formula / Rule: Quotient rule: d/dx (u/v) = [v (du/dx) − u (dv/dx)] / v² · Meaning:
- Formula / Rule: Second derivative: d²y/dx² = d/dx (dy/dx) · Meaning: Measures rate of change of the slope.
Common mistakes
- Forgetting to check that f(a) is defined when testing continuity → always verify the function value exists at the point.
- Assuming continuity guarantees differentiability → remember f(x) = |x| is continuous at 0 but has no derivative there.
- Omitting the inner derivative in the chain rule → when differentiating sin(3x), the answer is 3 cos(3x), not cos(3x).
- Treating y as a constant in implicit differentiation → every time you differentiate a term containing y, multiply by dy/dx.
- Confusing the Mean Value Theorem with Rolle's theorem → MVT applies whether or not f(a) = f(b); Rolle's requires f(a) = f(b).
Quick revision
- Continuous at a means the limit equals the function value there.
- Differentiable at a means the limit of the difference quotient exists; this forces continuity but not vice versa.
- Chain rule: multiply the outer derivative by the inner derivative.
- Implicit differentiation: differentiate the whole equation, then isolate dy/dx.
- Logarithmic differentiation turns products and powers into sums, simplifying differentiation.
- MVT guarantees at least one point where instantaneous rate equals average rate over an interval.