Derivatives and integrals — we compute them and explain what they mean
Function f(x)
Derivative
The derivative f′(x) shows how fast f(x) changes at every point: it is the “growth speed” of the function. If f(x) is the distance travelled, then f′(x) is the speed on the speedometer.
f″(x) is the second derivative: the rate of change of the speed itself (in the car example — the acceleration).
The number f′(x₀) is the slope of the tangent line — the line that hugs the graph at x₀. Positive slope — the function grows, negative — it falls, zero — a peak or a valley.
Definite integral
The integral ∫ f(x) from a to b is the area between the graph and the X axis. If f(x) is speed, the integral is the total distance travelled. Parts below the X axis count with a minus sign.
📚 Theory: what a derivative is
The derivative is the central notion of calculus. Formally it is the limit of the ratio of increments: f′(x) = lim (f(x+h) − f(x))/h as h → 0. Sounds scary, but the meaning is simple: we look at how much the function changed over a tiny step, divide by the step length — and get the rate of change.
Example: f(x) = x². At x = 3 the derivative is f′(3) = 6. It means: near x = 3 the function grows at a rate of “6 units of y per 1 unit of x”. Plot x² in our grapher — and you will see the parabola indeed climbs steeply there.
Differentiation is simply the process of finding the derivative. It follows rules: the derivative of a sum is the sum of derivatives, a product uses the product rule (u·v)′ = u′·v + u·v′, and nested functions use the chain rule: differentiate the outer function first, then multiply by the derivative of the inner one.
Where the derivative is zero, the function “freezes” for a moment — that is where the peaks and valleys of the graph are (extrema). This is how derivatives help find maxima and minima — from a ball’s trajectory to a company’s profit.
📚 Theory: what an integral is
The definite integral ∫ₐᵇ f(x) dx is the area of the figure between the graph of f(x) and the X axis on the segment from a to b. Imagine slicing the figure into thousands of narrow vertical strips: the area of each is ≈ f(x)·(strip width), and the integral is the sum of all strips as the slicing becomes infinitely fine.
Everyday meaning: if f(t) is your speed at time t, then the integral of speed over time is the whole distance travelled. That is exactly how an odometer “integrates” the speedometer.
Integration and differentiation are mutually inverse operations (that is the Newton–Leibniz formula, also known as the fundamental theorem of calculus): the integral of a derivative returns the original function. Differentiate x² to get 2x — and the integral of 2x from 0 to 3 gives 9 = 3² again.
Our calculator computes the integral numerically: it splits the segment into small pieces and carefully sums them up (Simpson’s method). So it can handle even functions that have no “nice” antiderivative.