Calculus studies change and accumulation. These guides explain limits, derivatives, integrals and series with worked examples and the reasoning that makes each rule work.
Each calculus idea rests on the one before it. Limits define the derivative, derivatives give the rules for rates of change, integrals reverse differentiation and measure accumulated area, and series approximate functions with polynomials. The guides in this section follow that path.
Alongside the rules you will find the reasons behind them, fully worked examples, and the checks that catch most errors: differentiate an antiderivative to confirm it, or compare a Taylor polynomial with the function at a nearby point. The derivative, integral, limit and series calculators show matching step-by-step work.
Calculus also rewards careful notation. Writing in front of what you are differentiating, keeping the on every indefinite integral and stating the interval for every series are small habits that prevent most lost marks, and the worked examples model each of them.
Students starting out should read the guide to limits, then what a derivative is and the derivative rules, before moving on to integrals and series.
Calculus
A limit describes the value a function approaches as its input approaches some number, whether or not the function actually reaches that value. Limits are the foundation of derivatives, integrals and infinite series, so evaluating them confidently is the first real skill of calculus.
Calculus
The derivative of a function measures how fast its output changes as its input changes, at a single instant. Graphically it is the slope of the tangent line; algebraically it is the limit of average rates of change over smaller and smaller intervals.
Calculus
Computing derivatives from the limit definition is slow. A small set of rules, each proved once from that definition, lets you differentiate almost any formula in a few lines. This guide explains each rule, why it works and how to combine them.
Calculus
An integral adds up infinitely many tiny pieces to find a total: an area, a distance, a volume or any accumulated quantity. The Fundamental Theorem of Calculus links this adding-up process to antiderivatives, which is what makes integrals practical to compute.
Calculus
An indefinite integral is a family of functions, the antiderivatives, written with a constant $+ C$. A definite integral has limits of integration and produces a single number, a signed area or accumulated total. The Fundamental Theorem of Calculus lets you compute the second from the first.
Calculus
Implicit differentiation finds dy/dx for curves defined by an equation in x and y, without first solving for y. You differentiate both sides with respect to x, treat y as a function of x so the chain rule applies, and then solve for dy/dx.
Calculus
A Taylor series rewrites a function as an infinite polynomial built from its derivatives at a single point. Cutting the series off after a few terms gives a Taylor polynomial, an approximation that is easy to compute and whose error can be bounded.
Calculus
An infinite series converges when its partial sums approach a finite number. Rather than computing that number, convergence tests decide whether it exists, each test suited to a particular shape of term. Knowing which test fits which series is the real skill.