The Product Rule

What to observe

1. Press Watch it grow and watch the edges move. The right edge slides out and paints a strip of area v·Δu: the old height, times how far it travelled. The top edge slides up and paints u·Δv the same way, across the old width. The little square where they meet belongs to neither, so it is a third piece all of its own.
2. That corner is Δu·Δv, and it is the only piece that dies. Halve h and each strip roughly halves, but the corner quarters, because it is short in two directions at once. Watch the big percentage collapse as you shrink h.
3. So the area gains one strip for each edge, and that is exactlyu′v + uv′. Divided by h, the bars on the right show each piece converging: the two strips onto their limits, the corner onto zero.
4. Now you can see why (uv)′ is not u′v′. The product of the two growths is only that doomed corner, the smallest piece on screen. The real growth is in the strips, each one an entire side times a small motion.
5. Notice the strips are usually lopsided. A long side paired with a slow growth can still out-paint a short side growing fast, which is precisely what u′v + uv′ keeps track of.
6. Pick x · (1.5+sin x) and push x past 1.57, the crest of v. Now the top edge slides down: the strip is scraped off rather than painted on, drawn dashed, and the corner flips sign with it. The rule never assumed the pieces were positive.