Limits and Derivatives

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Intuitive Limits and One-Sided Limits

Part of: Limits

Limits track nearby values, never the point itself; left and right approaches must agree; piecewise jumps split them; tables guess the trend.

Why it matters. Agreement decides existence. Approach beats substitution.

Formulas

Limit exists only if left limit = right limit

Key words

Limit: : Value f nears as x closes in, skipping the point itself.
Left-hand limit: : Approach checked from values below the point.
Right-hand limit: : Approach checked from values above the point.
Piecewise function: : Split definition that can break left-right agreement.
Limit does not exist: : No single value both sides agree on.

Try it in two minutes

Approach from both sides.
Compare the trends.
Declare only on agreement.

Try at home. Table x² near x = 3.

Algebra of Limits

Part of: Limits

Sums, products and quotients pass through limits; powers follow suit; (xⁿ−aⁿ)/(x−a) tends to n-a-to-n-minus-one; sin-x-over-x tends to one; (eˣ−1)/x tends to one.

Why it matters. Algebra splits hard limits. Standards anchor trig and exp.

Formulas

Limit of sum = sum of limits, when both exist
lim (x^n - a^n)/(x - a) = n x a^(n - 1)
lim sin x / x = 1 as x -> 0
lim (e^x - 1)/x = 1 as x -> 0

Key words

Limit laws: : Sums, products and quotients pass through limits.
Power quotient rule: : Standard result for power-difference quotients.
Sine standard limit: : Squeeze result forcing sin x over x to one.
Exponential limit: : Growth result forcing (e^x - 1) over x to one.
Indeterminate form: : Factor whose limit is unknown until simplified.

Try it in two minutes

Split by algebra.
Match a standard form.
Substitute the point.

Try at home. Verify sin-small-angle ratio.

Derivative from First Principle

Part of: Derivatives

Difference quotient over h tending zero defines f-prime; xⁿ yields n-x-to-n-minus-one; constants vanish; linear shifts scale simply.

Why it matters. First principle grounds every rule. Difference quotients measure rate.

Formulas

fprime(x) = limit [f(x + h) - f(x)] / h as h -> 0
d(x^n)/dx = n x x^(n - 1)
d(constant)/dx = 0

Key words

Difference quotient: : Slope limit of the chord over shrinking h.
First principle: : Limit defining the derivative at a point.
Derivative fprime: : Instant rate of change of f at x.
Power result: : Power result read directly from first principle.
Constant rule: : Flat lines whose slope limit is zero.

Try it in two minutes

Form f(x+h) minus f(x).
Divide by h.
Let h tend zero.

Try at home. Derive x² from scratch.

Rules of Differentiation

Part of: Derivatives

Power rule drops exponents down; sums split termwise; products cross-differentiate; quotients subtract-cross over squared denominator; sin goes cos, cos goes minus-sin.

Why it matters. Rules replace long limits. Cross patterns prevent slips.

Formulas

d(x^n)/dx = n x x^(n - 1)
(u + v)prime = uprime + vprime
(uv)prime = uprime v + u vprime
(u/v)prime = (uprime v - u vprime) / v^2
d(sin x)/dx = cos x and d(cos x)/dx = -sin x

Key words

Power rule: : Exponent drops down as multiplier of one power less.
Sum rule: : Derivatives split across plus and minus termwise.
Product rule: : Cross-differentiation of a product pair.
Quotient rule: : Subtract-cross over squared denominator.
Chain rule: : Outer rate times inner rate for nested functions.

Try it in two minutes

Drop the exponent.
Split sums.
Cross for products.

Try at home. Differentiate 3x⁴ + 2x.

What your child will practise

Algebra of Limits: Ocean Puzzle (Clue Puzzle)
Algebra of Limits: Ocean Quick Fire (Quick Fire)
Derivative from First Principle: Jungle Puzzle (Clue Puzzle)
Intuitive Limits and One-Sided Limits: Candy Quest (Flip and Match)
Rules of Differentiation: Desert Quick Fire (Quick Fire)

Quiz: Algebra of Limits (8 questions)
Quiz: Derivative from First Principle (8 questions)
Quiz: Intuitive Limits and One-Sided Limits (8 questions)
Quiz: Rules of Differentiation (8 questions)

Limits and Derivatives

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Intuitive Limits and One-Sided Limits

Part of: Limits

Limits track nearby values, never the point itself; left and right approaches must agree; piecewise jumps split them; tables guess the trend.

Why it matters. Agreement decides existence. Approach beats substitution.

Formulas

Limit exists only if left limit = right limit

Key words

Limit: : Value f nears as x closes in, skipping the point itself.
Left-hand limit: : Approach checked from values below the point.
Right-hand limit: : Approach checked from values above the point.
Piecewise function: : Split definition that can break left-right agreement.
Limit does not exist: : No single value both sides agree on.

Try it in two minutes

Approach from both sides.
Compare the trends.
Declare only on agreement.

Try at home. Table x² near x = 3.

Algebra of Limits

Part of: Limits

Sums, products and quotients pass through limits; powers follow suit; (xⁿ−aⁿ)/(x−a) tends to n-a-to-n-minus-one; sin-x-over-x tends to one; (eˣ−1)/x tends to one.

Why it matters. Algebra splits hard limits. Standards anchor trig and exp.

Formulas

Limit of sum = sum of limits, when both exist
lim (x^n - a^n)/(x - a) = n x a^(n - 1)
lim sin x / x = 1 as x -> 0
lim (e^x - 1)/x = 1 as x -> 0

Key words

Limit laws: : Sums, products and quotients pass through limits.
Power quotient rule: : Standard result for power-difference quotients.
Sine standard limit: : Squeeze result forcing sin x over x to one.
Exponential limit: : Growth result forcing (e^x - 1) over x to one.
Indeterminate form: : Factor whose limit is unknown until simplified.

Try it in two minutes

Split by algebra.
Match a standard form.
Substitute the point.

Try at home. Verify sin-small-angle ratio.

Derivative from First Principle

Part of: Derivatives

Difference quotient over h tending zero defines f-prime; xⁿ yields n-x-to-n-minus-one; constants vanish; linear shifts scale simply.

Why it matters. First principle grounds every rule. Difference quotients measure rate.

Formulas

fprime(x) = limit [f(x + h) - f(x)] / h as h -> 0
d(x^n)/dx = n x x^(n - 1)
d(constant)/dx = 0

Key words

Difference quotient: : Slope limit of the chord over shrinking h.
First principle: : Limit defining the derivative at a point.
Derivative fprime: : Instant rate of change of f at x.
Power result: : Power result read directly from first principle.
Constant rule: : Flat lines whose slope limit is zero.

Try it in two minutes

Form f(x+h) minus f(x).
Divide by h.
Let h tend zero.

Try at home. Derive x² from scratch.

Rules of Differentiation

Part of: Derivatives

Power rule drops exponents down; sums split termwise; products cross-differentiate; quotients subtract-cross over squared denominator; sin goes cos, cos goes minus-sin.

Why it matters. Rules replace long limits. Cross patterns prevent slips.

Formulas

d(x^n)/dx = n x x^(n - 1)
(u + v)prime = uprime + vprime
(uv)prime = uprime v + u vprime
(u/v)prime = (uprime v - u vprime) / v^2
d(sin x)/dx = cos x and d(cos x)/dx = -sin x

Key words

Power rule: : Exponent drops down as multiplier of one power less.
Sum rule: : Derivatives split across plus and minus termwise.
Product rule: : Cross-differentiation of a product pair.
Quotient rule: : Subtract-cross over squared denominator.
Chain rule: : Outer rate times inner rate for nested functions.

Try it in two minutes

Drop the exponent.
Split sums.
Cross for products.

Try at home. Differentiate 3x⁴ + 2x.

What your child will practise

Algebra of Limits: Ocean Puzzle (Clue Puzzle)
Algebra of Limits: Ocean Quick Fire (Quick Fire)
Derivative from First Principle: Jungle Puzzle (Clue Puzzle)
Intuitive Limits and One-Sided Limits: Candy Quest (Flip and Match)
Rules of Differentiation: Desert Quick Fire (Quick Fire)

Quiz: Algebra of Limits (8 questions)
Quiz: Derivative from First Principle (8 questions)
Quiz: Intuitive Limits and One-Sided Limits (8 questions)
Quiz: Rules of Differentiation (8 questions)