Derivative & Integral Visualizer

Enter a function and see it alongside its derivative and integral. Drag a point along the curve to inspect the tangent line slope and the accumulated area. All computations run in your browser.

Enter a function

f(x) =

Presets

Reference Guide

The Derivative

The derivative $f'(x)$ gives the instantaneous rate of change of $f(x)$ at every point. Geometrically, it equals the slope of the tangent line to the curve.

Definition

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Notation

f'(x)

\frac{dy}{dx}

\frac{df}{dx}

all represent the same thing.

Where $f'(x) > 0$ , the original function is increasing. Where $f'(x) < 0$ , it is decreasing. Points where $f'(x) = 0$ are candidates for local extrema.

The Integral

The integral $F(x) = \int_0^x f(t)\,dt$ gives the accumulated (signed) area under $f$ from 0 to $x$ .

Fundamental Theorem of Calculus

\frac{d}{dx}\int_0^x f(t)\,dt = f(x)

This connects differentiation and integration. The derivative of the area function returns the original function.

When $f(x) > 0$ , $F(x)$ increases. When $f(x) < 0$ , $F(x)$ decreases. The rate of change of the area equals the height of the curve.

Differentiation Rules

Power rule

\frac{d}{dx} x^n = n x^{n-1}

Product rule

\frac{d}{dx}[f \cdot g] = f' \cdot g + f \cdot g'

Quotient rule

\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f' g - f g'}{g^2}

Chain rule

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

Numerical vs Symbolic

Symbolic differentiation applies algebraic rules to produce an exact formula for

f'(x)

. For example,

\frac{d}{dx}(x^3) = 3x^2

is exact.

Numerical integration approximates the area under the curve using Simpson's rule, dividing the interval into many small pieces and summing weighted function values.

Why numerical? Some functions, like

e^{-x^2}

, have no closed-form antiderivative. Numerical methods give accurate approximations even when an exact formula does not exist.

This tool uses symbolic differentiation for $f'(x)$ and Simpson's rule numerical integration for $F(x)$ .