The Sum Rule

What to observe

1. Press Watch the step and keep your eyes on the tower. The bottom block pushes up by Δu, carrying the whole upper block with it. Then the upper block grows by Δv on its own. The top moves by both amounts, one after the other, and nothing else happens.
2. That is the entire rule. Divide those two rises by h and they become slopes: (u+v)′ = u′ + v′. The sentence under the equation says the same thing in words, with the live numbers filled in.
3. Look at the three triangles at the bottom right. They share the same base h, and the tall one's height is literally the other two stacked. Two slopes, stacked, make the third slope.
4. Shrink h. Everything shrinks together and the ratios never budge. Unlike the product rule, nothing here has to be thrown away in the limit: Δ(u+v) = Δu + Δv is exactly true at every h, not just as h approaches zero.
5. Try x² and 2.4−x, where v falls while u climbs. The upper block shrinks, its rise is negative, and the tower still ends up exactly as tall as the two rises say. Addition never cared about signs.
6. Ask why this feels obvious, because the answer matters. Stacking never lets the two interact: how tall u is does not change how fast v grows. Multiplying does let them interact, and that interaction is exactly the extra term in the product rule.