Differentiation vs. Integration: Understanding Calculus Through Visualization

Differentiation and Integration are the two core operations in calculus. They are inverse processes, linked by the Fundamental Theorem of Calculus.

The animation visually contrasts them using the same curve shape on two sides.

1. Differentiation (Left Side)
Differentiation finds the instantaneous rate of change (slope) of a function at a specific point.

Formula (as shown in the animation):

$$f'(x_0) = lim_{h to 0} frac{f(x_0 + h) – f(x_0)}{h}$$

This is the definition of the derivative.
As $ h $ (a tiny horizontal change) approaches 0, the secant line (green in the animation) becomes the tangent line.
Visually: The animation zooms in or shrinks $ h $ to show how the average slope over a small interval becomes the exact slope at $ x_0 $.