Derivative Visualizer 2D: Exploring Numeric vs Analytic Slopes

What Derivatives Are

A derivative measures how a function changes as its input changes. It's essentially the slope of the tangent line to a curve at any given point.

In mathematical terms, if we have a function f(x), the derivative f'(x) represents the rate of change of f with respect to x.

Analytic vs Numeric Derivatives

The analytic derivative is derived algebraically from the original function. For example, for f(x) = x^2, its derivative is f'(x) = 2x.

A numeric derivative approximates this by using small differences in the function values over a short interval h, such as (f(x + h) - f(x)) / h.

Visualizing Derivatives

The visualizer allows you to see how these derivatives are calculated and their accuracy. By dragging x0 on the curve or its slider, you can observe the tangent line at that point.

Shrinking h makes the secant line approach the tangent more closely, illustrating the approximation error between numeric and analytic methods.

Concavity and Second Derivatives

The visualizer also estimates concavity via f''(x0), which is the second derivative of the function at x0. This helps in understanding how the first derivative itself changes.

For a quadratic function, the second derivative is constant, reflecting the linear nature of its rate of change.

Frequently asked questions

Try it live

Everything above runs in your browser — open Derivative Visualizer 2D: Numeric vs Analytic Slope and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.