Both Sides Move, So Two Terms Come Out

You'll learnwhy the derivative of a product has exactly two terms, how to assemble them without losing one, and when expanding first beats using the rule at all.

Differentiation splits a sum, so it is a fair guess that it splits a product too. Take f(x) = x and g(x) = x. Then f · g = x², whose derivative is 2x — while f′ · g′ = 1 · 1 = 1. Two different functions, so no rule of that shape exists, and one example is enough to retire it for good. What replaces it is a picture. A product is an area: f · g is a rectangle with sides f and g. Push x forward a little and the rectangle gains an L-shaped strip — one piece because the width grew, one piece because the height grew, and a small corner where both grew at once. Divide the whole gain by the step and shrink it. The two strips survive as g f′ and f g′, the corner dies, and there is the rule: two terms, each pairing one derivative with the other factor left exactly as it was.

8 STEPS · 6 QUICK CHECKS · TWO TERMS, NEVER ONE · EACH DERIVATIVE PAIRS WITH THE OTHER FACTOR · v1

(f g)′ = f′g + fg′ · never f′g′

Before you start

What you're looking at

The product (f g)(x) = (x + 1)x², drawn as a rectangle whose width is f(x) = x + 1 and whose height is g(x) = x². Its area is the product.

The question

Step x forward a little. How much new area appears — and which parts of the rectangle does it appear in?

Watch for

The L-shaped strip: one piece from the width growing, one from the height growing, and a corner that shrinks away faster than either of them.

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