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 is the better road.

A product is an area: a rectangle whose width is one function and whose height is the other. Push x forward and both sides move at once. How much new area appears, and where does it show up? Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · TWO TERMS, NEVER ONE · EACH DERIVATIVE PAIRS WITH THE OTHER FACTOR · v2

(f g)′ = f′g + fg′ · never f′g′

Before you start

What you're looking at

A rectangle whose width is f(x) = x + 1 and whose height is g(x) = x². Its area is the product f · g.

The question

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

Step 1 / 12