Product rule

“The derivative of a product is NOT the product of derivatives.”

d/dx(uv) = u'v + uv'.

(uv)'=u'v+uv'

Memory trick: First×d(second) + second×d(first).

Chain rule

“For sin(2x) it's cos(2x)×2 — never forget the ×2.”

Differentiate the outer function, then multiply by the inner derivative.

dy/dx=dy/du·du/dx

Memory trick: Outer' × inner'.

Maxima and minima

“Businesses use this to squeeze out maximum profit or minimum cost.”

At a maximum or minimum, f'(x)=0; use f''(x) to tell which (f''<0 max, f''>0 min).

Memory trick: f'=0 is only a candidate — check f''.

Increasing and decreasing

“The sign of the derivative tells you which way the curve is heading.”

Where f'(x) > 0 a function rises; where f'(x) < 0 it falls.

Memory trick: f'>0 up, f'<0 down.

Integration as antiderivative

“Undoing a derivative always loses the constant — hence +C.”

Integration reverses differentiation; indefinite integrals carry a +C.

∫xⁿ dx=x^(n+1)/(n+1)+C

Memory trick: Never forget +C.

Definite integral as area

“Adding infinitely many thin strips gives the exact area of a curved region.”

The definite integral gives the (signed) area under a curve between two limits.

Memory trick: Area below axis counts negative.

Fundamental theorem of calculus

“It's the bridge that ties the two halves of calculus together.”

Differentiation and integration are inverse operations.

Memory trick: Integrate then differentiate → back to start.

Differential equation

“Population growth, cooling coffee and radioactive decay are all differential equations.”

An equation involving a function and its derivatives.

Memory trick: Relates a function to its rate of change.

Dot product

“The dot product measures how much two vectors point the same way.”

a·b = |a||b|cosθ gives a scalar; it's zero when vectors are perpendicular.

a·b=|a||b|cosθ

Memory trick: Dot → scalar; zero → perpendicular.

Cross product

“The cross product points out of the plane, right-hand-rule style.”

a×b is a vector perpendicular to both, with magnitude |a||b|sinθ.

|a×b|=|a||b|sinθ

Memory trick: Cross → vector; zero → parallel.

Matrix

“Computer graphics rotate and scale every image using matrices.”

A rectangular array of numbers used to store and transform data.

Memory trick: Rows × columns.

Matrix multiplication is non-commutative

“Order matters: rotating then scaling differs from scaling then rotating.”

In general AB ≠ BA for matrices.

Memory trick: AB ≠ BA.

Determinant

“A zero determinant means a transformation squashes space flat.”

A single number from a square matrix; if it's zero, the matrix has no inverse.

Memory trick: det = 0 → no inverse.

Probability basics

“From weather to cricket, uncertainty is measured on this 0-to-1 scale.”

Probability = favourable outcomes ÷ total outcomes, always between 0 and 1.

P=favourable/total

Memory trick: Never below 0 or above 1.

Independent vs mutually exclusive

“Mutually exclusive events are actually dependent — one rules the other out.”

Independent events don't affect each other; mutually exclusive events can't both happen.

Memory trick: Exclusive ≠ independent.

Conditional probability

“Medical tests are interpreted with conditional probability.”

P(A|B) = P(A∩B)/P(B) is the chance of A given B has occurred.

P(A|B)=P(A∩B)/P(B)

Memory trick: Given B, rescale by P(B).

Bayes' theorem

“Spam filters use Bayes to keep learning what junk looks like.”

Updates a probability when new evidence arrives, flipping P(A|B) into P(B|A).

Memory trick: Reverse the condition correctly.

Mean, median, mode

“A few billionaires can drag the mean income far above the typical (median) person.”

Mean is the average, median the middle value, mode the most frequent.

Memory trick: Skewed data → trust the median.

Standard deviation

“Two classes can share an average yet differ wildly in consistency.”

A measure of how spread out data is around the mean.

Memory trick: Bigger SD = more spread.

Set

“Your phone's contact list is a set — no duplicates allowed.”

A well-defined collection of distinct objects.

Memory trick: Elements are unique.

Union and intersection

“Union is 'either'; intersection is 'both' — like overlapping friend circles.”

Union (A∪B) gathers all elements of either set; intersection (A∩B) keeps only the common ones.

Memory trick: ∪ = OR, ∩ = AND.

Subset

“A team is a subset of the whole class.”

A is a subset of B if every element of A is in B.

A⊆B

Memory trick: All of A sits inside B.

De Morgan's laws

“Negating an 'or' flips it to an 'and' — a rule programmers use daily.”

(A∪B)' = A'∩B' and (A∩B)' = A'∪B'.

Memory trick: Complement flips ∪ and ∩.

Power set

“Three toppings give 8 possible pizzas — that's the power set.”

The set of all subsets of a set; a set with n elements has 2ⁿ subsets.

|P(A)|=2ⁿ

Memory trick: n elements → 2ⁿ subsets.