Visualizing the chain rule and product rule | Chapter 4, Essence of calculus

Stacked heights, an expanding rectangle and successive number-line mappings explain how local increments lead to sum, product and composition rules. This video visualizes the chain rule of calculus using a three-tiered number line model to track infinitesimal changes. By starting with a concrete value $x = 1.5$ and applying a tiny nudge $dx$, it demonstrates how this change propagates through nested functions like squaring and sine. The animation shows that the total change in the composite function is the product of the local scaling factors at each stage. This leads to the general formula for the derivative of a composition $g(h(x))$. The video concludes by emphasizing that understanding these mechanical rules requires active practice rather than passive viewing.

Before you watch

• Derivatives and difference quotients
• Operations on functions
• Algebraic expansion