Single-variable calculus from first principles
When two functions are multiplied together, you can't just multiply their derivatives. That's a tempting shortcut, and a wrong one. The right rule accounts for the fact that both factors are changing at once.
Picture a rectangle whose width is f and height is g; its area is f·g. If both sides grow a little, the area grows on two fronts: a strip from the wider width, plus a strip from the taller height. That's why the answer has two terms, not one.
Picture a rectangular garden whose width and height are both being extended at once. The new area isn't just one strip, you gain a strip along the longer width and a strip along the taller height. That's why the product rule has two terms: when two changing quantities multiply, each one's growth contributes its own slice to the total.