A friendly introduction to calculus
Calculus is two ideas — differentiation and integration — connected by the most beautiful theorem in maths.
What problem does calculus solve?
Algebra is great at handling fixed quantities. The moment something changes — a falling apple, a bank balance growing with interest, the temperature of a cup of coffee — algebra runs out of road. Calculus picks up where algebra stops. It gives us a precise way to talk about rates of change (differentiation) and about accumulated quantities (integration).
Limits: the language of "approaches"
Imagine zooming into the graph of f(x) = x² around the point x = 2. As you zoom in, the curve looks more and more like a straight line. The slope of that line as you zoom in further and further is what we call the limit — a value that the function approaches but may or may not actually reach. Limits are the technical machinery underneath everything else in calculus.
The derivative is a rate of change
The derivative of f at x, written f′(x) or df/dx, is the slope of the curve at that point. Formally, it is a limit:
f′(x) = limh→0 (f(x + h) − f(x)) / h
You almost never compute it from this definition. Instead you use the rules: the power rule, the product rule, the quotient rule, and the chain rule. They cover almost every function you will meet in school. The derivative calculator applies all four automatically.
What derivatives tell you
If f(x) is the position of a moving object at time x, then f′(x) is its velocity. If f(x) is your bank balance, f′(x) is the rate at which money is flowing in or out. If f(x) is a cost function, then setting f′(x) = 0 finds the points where cost is at a minimum or maximum — the foundation of every optimisation problem in economics and engineering.
The integral is an accumulator
If derivatives slice things into instantaneous rates, integrals add things up. The definite integral ∫ₐᵇ f(x) dx is the total area under the curve f from x = a to x = b. It is also the limit of an infinite Riemann sum: chop the interval into n tiny strips of width Δx, multiply each by f(x), add them up, then let n → ∞.
The integral has the same connection to its setup that the derivative has to its: where derivatives describe instantaneous behaviour, integrals describe cumulative behaviour. If f(x) is your speed at time x, the integral from a to b is the distance you covered between times a and b.
The fundamental theorem of calculus
The bridge between differentiation and integration is the most important result in elementary calculus. It says, roughly, that integration and differentiation are inverse operations:
∫ₐᵇ f′(x) dx = f(b) − f(a)
So if you know an antiderivative — a function whose derivative is f — you can compute any definite integral by evaluating the antiderivative at the two endpoints and subtracting. That single fact is what makes calculus practical, because differentiation has clean rules but direct integration of a Riemann sum is painful.
What you should learn next
Once the rules feel comfortable, learn integration techniques (substitution and integration by parts), then move to multivariable calculus (partial derivatives, double integrals) and differential equations. But the single-variable picture you have now is already enough to do most of the calculus that physics, statistics, and machine learning rely on.