The Product Rule

When two quantities that both depend on x get multiplied together, the product changes for two separate reasons: the first factor is changing, and the second factor is changing. You cannot just multiply the two derivatives together, because that would only catch one source of change and would get the units and the sizes wrong. The product rule accounts for both: (uv)' = u'v + uv'. Each term freezes one factor at its current value and lets the other one vary.

Picture a rectangle with width u(x) and height v(x), so its area is u(x)v(x). Nudge x forward a little. The width grows by a small amount and the height grows by a small amount, so the area picks up a sliver from the width growing (that sliver has height v) and a sliver from the height growing (that sliver has width u). There is also a tiny corner piece from both changing at once, but it shrinks away faster than the other two pieces as the nudge gets small, so it vanishes in the limit. That leaves exactly u'v + uv'.

This rule is what makes differentiation fast: once you have it, you never need to expand a product or go back to the limit definition just because two functions got multiplied. It sits alongside the sum rule and the chain rule as one of the tools that turns derivatives of complicated expressions into bookkeeping, and it will reappear inside the quotient rule and in later related-rates and optimization problems where two changing quantities are multiplied together.