Derivative: Limit Approximation
Discover the geometric meaning of the derivative by visualizing how a secant line becomes a tangent line as the limit of Δx approaches zero.
About this simulation
- What
- An interactive Math simulation of Derivative: Limit Approximation.
- Who
- Designed for AP, IB, and A‑Level Math students.
- How
- Runs in any modern browser — drag, adjust, and explore in real time.
Updated 2026-04-01
Key Concepts
Secant Line (Average Rate)
A line intersecting two points on a curve. Its slope represents the average rate of change over the interval Δx.
Tangent Line (Instantaneous Rate)
A line that just touches the curve at a single point, representing the instantaneous rate of change at that exact point.
Limit Approximation
As the distance Δx between two points approaches zero, the secant line smoothly approaches the tangent line.
Derivatives As Slopes
In calculus, the derivative of a function measures how the output value changes relative to its input. Geometrically, this is represented by the slope of the tangent line at a specific point.
The rigorous definition of a derivative is based on limits: we compute the slope of a secant line between points x and x+Δx, and then observe what happens as Δx shrinks to exactly 0.
Use the slider above to drag Δx towards 0. Notice how the triangle representing the differences in y and x becomes infinitely small, forcing the intersecting secant line to morph into the tangent line.