What about a product of two functions, like x^2 \sin(x)? A tempting guess is to differentiate each factor and multiply — but that is wrong. The slope of a product is not the product of the slopes.

The correct rule keeps both factors in play at every step. For u(x) and v(x):

Read it as: differentiate the first, leave the second; then leave the first, differentiate the second — and add. Each factor takes a turn being the one that changes.

Why: a growing rectangle

Picture u and v as the two sides of a rectangle, so the product uv is its area. As x ticks forward, both sides grow a little: u grows by du and v by dv. How much new area appears?

Two thin strips: one along the top of width u and thickness dv (area u\,dv), and one along the side of height v and thickness du (area v\,du). The tiny corner square du\,dv is negligible. So the area grows by u\,dv + v\,du — which is exactly the product rule. Drag the slider to grow the sides and watch the two strips appear.

A worked example

Differentiate f(x) = (x^2)(x^3) with the product rule. Take u = x^2 and v = x^3, so by the power rule u' = 2x and v' = 3x^2:

That's a reassuring check: x^2 \cdot x^3 = x^5, and the power rule gives 5x^4 directly — the product rule agrees. For factors you can't simply multiply out (like x^2 \sin x), the product rule is the only way through:

Watch Sal explain it

The two-strip picture is the heart of it; the rest is bookkeeping. Here is the product rule introduced and applied: