Simplifying Complex Products

Calculus 1 · Axiom Academy

The derivative of a product is not the product of the derivatives — here is the rule that actually works. The tempting shortcut that quietly fails You can differentiate a sum term by term: (f+g)′ = f′ + g′ . So when you first meet a product like , it is almost irresistible to guess that products work the same way — just differentiate each piece and multiply: . It is a clean idea. It is also wrong, and it is wrong by a lot. Watch it on a product you can check by hand. Take and , so the product is . At the real slope of that curve is . The naive guess draws a line that is far too shallow. The shallow line is what "multiply the derivatives" predicts. The steeper line is the slope the curve actually has — they are not the same line. It is not a fluke at one point — the guess misses everywhere Maybe was unlucky. Slide the focus point along the same product and compare, at every position, the naive guess against the true slope . The two readouts never agree — and the gap between them only grows. Whatever x you pick, "multiply the derivatives" reads too low. So what is the missing slope made of? A product is an area — grow it and watch the pieces Picture the product as the area of a rectangle with one side and the other side . Nudge forward by a small step : one side grows by and the other by . Drag the step down toward zero and see exactly which pieces of new area survive. As x → 0 the corner per step collapses to 0, leaving only the two strips: that is exactly f′g + fg′.

This is the written version of the interactive lesson above. See the full Calculus 1 course.