product rule

Suppose two things are both growing at once — say a rectangle whose width and height are each stretching as time passes. How fast is its area growing? You might guess: just multiply the two growth rates. But that misses most of the answer. The product rule is the correct recipe for the rate of change of a product of two functions, and the rectangle picture shows exactly why the naive guess fails.

The rule is (fg)' = f'g + fg'. In words: the derivative of a product is (the derivative of the first times the second) plus (the first times the derivative of the second). For the growing rectangle of width f and height g, when both edges nudge outward the new area added comes in two strips — one from the width growing (rate f' across the full height g) and one from the height growing (rate g' across the full width f). Add those two strips and you get f'g + fg'.

The mistake to avoid is writing (fg)' = f'g' — that is simply wrong. You can see it with a quick test: let f(x) = x and g(x) = x, so fg = x^2. The product rule gives (x^2)' = 1*x + x*1 = 2x, which is correct. The bogus rule f'g' would give 1*1 = 1, which is not the derivative of x^2. The two strips are real; you can't drop one of them.