Topic
Calculus
The math of change. Derivatives describe rates, integrals add up tiny pieces, and limits hold both ideas together.
Two big ideas
Differential calculus asks: how fast is something changing? Integral calculus asks: if I add up infinitely many small things, what do I get? Surprisingly, they're inverses of each other.
Start here
What calculus is really about — and the one-sentence link between its two halves.
Limits & continuity
Sneak up on a value. The idea that makes every derivative and integral possible.
Derivatives
The slope of a curve at a point — what it means and when it exists.
Differentiation rules
The four moves that handle almost every derivative — plus the tricky cases.
Differentiation rules
Power, product, quotient and chain — the four moves you really need.
Power rule
d/dx(xⁿ) = nxⁿ⁻¹. The most-used differentiation rule.
Product rule
d/dx(uv) = u'v + uv'. For when two functions are multiplied.
Chain rule
Differentiate the outside, multiply by the derivative of the inside.
Implicit differentiation
When y is mixed with x — differentiate both sides and solve for dy/dx.
Trig derivatives proof
Why d/dx(sin x) = cos x — proved with limits and the squeeze theorem.
Applying derivatives
Use the derivative to find peaks, troughs, curvature — and to crack tricky limits.
Maxima & minima
Where a function peaks or bottoms out — set the derivative to zero.
Stationary points
Points where the derivative is zero — peaks, troughs, and saddles.
Inflection points
Where the curve changes its bend — concave up flips to concave down.
Concave up & down
Cup-up or cup-down — and what the second derivative tells you.
Second derivative
The derivative of the derivative — measures curvature and acceleration.
L'Hôpital's rule
When 0/0 or ∞/∞ shows up, differentiate top and bottom.
Integrals
Sum tiny strips to find an area — and turn antiderivatives into exact answers.
Integrals
Sum tiny strips to find an area — Riemann sums live.
Integration introduction
Why integration is just the reverse of differentiation.
Integration rules
The basic rules — power, sum, constant — that handle most integrals.
Definite integrals
From a to b — turn an antiderivative into an exact area.
Integration techniques
When the basic rules aren't enough — substitute, split, or approximate.
Applications of integrals
Use integration to measure curves and 3D shapes.
The big connection
Why differentiation and integration are inverses — the theorem that ties calculus together.
Differential equations
Equations where the unknown is a function — and a derivative is in the mix.
Differential equations
Equations where the unknown is a function — and a derivative is in there too.
Separation of variables
Get all the y's on one side, all the x's on the other, then integrate both.
First-order linear DE
dy/dx + P(x)y = Q(x) — the integrating-factor recipe.
Second-order DE
y'' + by' + cy = 0 — characteristic equations and three solution families.
Series
Build any nice function out of polynomials — or any periodic one out of sines.
Multivariable
Calculus when there's more than one input — slope one variable at a time.