Schooling · Class 11 · Mathematics · Limits and Derivatives · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Limits and Derivatives: 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.
This note's current text has no applied fact-check on record.
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
- The limit of f(x) as x approaches a…
- If direct substitution gives a form…
- Two important standard limits are
- The derivative of f(x) at x = a is…
- Geometrically, the derivative f'(a)…
- The derivative of a sum is the sum…
- Derivatives of basic functions…
- Formulas and facts
- Limit rules (assuming both limits…
- lim x
- Standard limits
- Derivative formulas
- d/dx (c) = 0
- d/dx (x^n) = n·x^(n−1)
- d/dx (sin x) = cos x
- Watch out for
- Substituting directly into 0/0 forms…
- Forgetting that (sin x)/x
- Writing d/dx (cos x) = sin x…
- Applying the product rule…
- Confusing the derivative…
What this chapter is about
This chapter introduces you to calculus, one of the most powerful tools in all of mathematics. Calculus studies how quantities change, and it begins with two fundamental concepts: limits and derivatives. A limit describes what value a function approaches as its input gets closer and closer to some number, even if the function is not defined exactly at that number. Understanding limits is essential because derivatives are built on them.
Key ideas
- The limit of f(x) as x approaches a, written lim x→a f(x), is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a, regardless of whether f(a) is defined.
- If direct substitution gives a form like 0/0, you must simplify the expression by factoring, rationalising, or using standard limit results before evaluating.
- Two important standard limits are: lim x→0 (sin x)/x = 1 (where x is in radians) and lim x→0 (1 − cos x)/x = 0.
- The derivative of f(x) at x = a is defined as lim h→0 [f(a + h) − f(a)]/h, provided this limit exists. This is called differentiation from first principles.
- Geometrically, the derivative f'(a) gives the slope of the tangent line to the curve y = f(x) at the point where x = a.
- The derivative of a sum is the sum of derivatives, and a constant factor can be taken outside the derivative: d/dx [c·f(x)] = c·f'(x).
- Derivatives of basic functions follow patterns: d/dx (x^n) = n·x^(n−1) for any real n, d/dx (sin x) = cos x, and d/dx (cos x) = −sin x.
Formulas and facts to remember
- Limit rules (assuming both limits exist):
- lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
- lim x→a [f(x) · g(x)] = lim x→a f(x) · lim x→a g(x)
- lim x→a [f(x)/g(x)] = [lim x→a f(x)] / [lim x→a g(x)], provided denominator is not zero
- Standard limits:
- lim x→a (x^n − a^n)/(x − a) = n·a^(n−1)
- lim x→0 (sin x)/x = 1 (x in radians)
- lim x→0 (tan x)/x = 1
- lim x→0 (1 − cos x)/x² = 1/2
- Derivative formulas:
- d/dx (c) = 0, where c is a constant
- d/dx (x^n) = n·x^(n−1)
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec²x
- Derivative rules:
- Sum rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
- Constant multiple: d/dx [c·f(x)] = c·f'(x)
- Product rule: d/dx [f(x)·g(x)] = f(x)·g'(x) + g(x)·f'(x)
- Quotient rule: d/dx [f(x)/g(x)] = [g(x)·f'(x) − f(x)·g'(x)] / [g(x)]²
Common mistakes
- Substituting directly into 0/0 forms without simplifying first → always check if the result is indeterminate, then factorise or rationalise before substituting.
- Forgetting that (sin x)/x → 1 only when x is in radians → always work in radians for calculus involving trigonometric limits.
- Writing d/dx (cos x) = sin x (missing the negative sign) → remember d/dx (cos x) = −sin x.
- Applying the product rule incorrectly as f'(x)·g'(x) → the correct form is f(x)·g'(x) + g(x)·f'(x).
- Confusing the derivative (instantaneous rate) with the average rate of change → the derivative uses a limit as h → 0, not a finite change.
Quick revision
- A limit describes the value a function approaches, not necessarily the value it reaches.
- lim x→0 (sin x)/x = 1 is fundamental; memorise it.
- Derivative = lim h→0 [f(x + h) − f(x)]/h = slope of tangent = instantaneous rate of change.
- Power rule: d/dx (x^n) = n·x^(n−1).
- Product and quotient rules let you differentiate combinations of functions.