Implicit differentiation

“It handles curves like circles that aren't neat y = f(x).”

Differentiate both sides treating y as a function of x, then solve for dy/dx.

Memory trick: Differentiate y, then isolate dy/dx.

Rolle's theorem

“If you return to the same height, somewhere your path was momentarily flat.”

If f is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then f'(c)=0 for some c.

Memory trick: Equal ends → a flat point.

Mean value theorem

“On any trip, at some instant your speed equalled your average speed.”

Somewhere on a smooth curve, the instantaneous slope equals the average slope.

Memory trick: Instant slope = average slope.

Integration by parts

“It trades a hard integral for an easier one, one factor at a time.”

∫u·dv = uv − ∫v·du.

∫u dv = uv − ∫v du

Memory trick: Choose u by ILATE.

Integration by substitution

“Spotting the right substitution is like finding the loose thread that unravels the knot.”

Replace a chunk with a new variable to simplify the integral.

Memory trick: Undo the chain rule.

Standard integral of 1/x

“The power rule breaks at n=−1 — and the logarithm rides to the rescue.”

∫(1/x)dx = ln|x| + C.

∫1/x dx=ln|x|+C

Memory trick: 1/x integrates to ln|x|.

Property of definite integrals

“This 'king' property often collapses a nasty integral in one line.”

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx.

Memory trick: Swap x → a+b−x.

Area between curves

“Integration measures the exact area of shapes no formula covers.”

The area between two curves is the integral of (top − bottom).

Memory trick: ∫(top − bottom).

Order and degree of a differential equation

“They classify a differential equation before you try to solve it.”

Order is the highest derivative present; degree is its power after clearing radicals.

Memory trick: Order = highest derivative.

Variable separable equations

“The simplest differential equations just ask you to sort x's and y's apart.”

If dy/dx = f(x)g(y), separate variables and integrate both sides.

Memory trick: Get x's and y's on opposite sides.

Vector addition (triangle law)

“Rowing across a river, your path is the resultant of boat and current.”

Two vectors add head-to-tail, the resultant closing the triangle.

Memory trick: Head-to-tail resultant.

Unit vector

“It strips away size and keeps only 'which way'.”

A vector of magnitude 1 giving pure direction, â = a/|a|.

â = a/|a|

Memory trick: Divide by its own length.

Scalar triple product

“If it's zero, the three vectors are coplanar.”

a·(b×c) gives the volume of the parallelepiped formed by three vectors.

[a b c]

Memory trick: Zero → coplanar vectors.

Projection of a vector

“It's the 'shadow' one vector casts along another.”

The projection of a on b is (a·b)/|b|.

(a·b)/|b|

Memory trick: Uses the dot product.

Direction cosines

“They pin down a line's direction in 3-D space.”

The cosines of the angles a line makes with the axes; they satisfy l²+m²+n²=1.

Memory trick: l²+m²+n² = 1.

Angle between two planes

“Two walls meet at the angle their perpendiculars make.”

Equals the angle between their normal vectors.

Memory trick: Compare the normals.

Transpose of a matrix

“Flip a matrix across its diagonal.”

Swap rows and columns to get Aᵀ.

Memory trick: Rows become columns.

Symmetric and skew-symmetric

“Any square matrix splits into a symmetric plus a skew-symmetric part.”

Symmetric: Aᵀ=A; skew-symmetric: Aᵀ=−A (with zero diagonal).

Memory trick: Aᵀ=A vs Aᵀ=−A.

Inverse of a matrix

“The inverse 'undoes' a matrix transformation — if the determinant isn't zero.”

A⁻¹ = adj(A)/det(A), and exists only when det(A) ≠ 0.

A⁻¹=adjA/detA

Memory trick: No inverse when det = 0.

Cramer's rule

“A neat determinant recipe for solving simultaneous equations.”

Solves linear systems using ratios of determinants.

Memory trick: Each unknown = ratio of determinants.

Area of a triangle by determinant

“A determinant of zero means the three points are collinear.”

The area from vertices is half the absolute value of a 3×3 coordinate determinant.

Memory trick: Zero area → collinear points.

Addition theorem of probability

“Subtract the overlap so you don't count it twice.”

P(A∪B) = P(A) + P(B) − P(A∩B).

P(A∪B)=P(A)+P(B)−P(A∩B)

Memory trick: Don't double-count the overlap.

Multiplication theorem

“Two fair coins both landing heads: ½ × ½ = ¼.”

For independent events, P(A∩B) = P(A)·P(B).

Memory trick: Independent → multiply.

Binomial distribution

“It predicts how many heads in 10 coin tosses, or defects in a batch.”

Models the number of successes in n independent yes/no trials with fixed probability p.

Memory trick: n trials, probability p each.