The Product Rule - Differentiating Products of Functions

Calculus 1 · Axiom Academy

How to differentiate two functions multiplied together — derived from the limit, seen as growing area, and run on real products. 1. The Statement: A Product Has Two Terms For two differentiable functions f(x) and g(x) , the derivative of their product is: Differentiate one factor at a time, keep the other, and add. The animation tests this on the concrete product f = x² , g = sin x . The naive guess f′ g′ gives one answer; the real rule gives a different one. They disagree — so the naive guess is wrong. — multiplying the two derivatives drops a whole term. (fg)' = f'g + fg' — two terms, each with exactly one derivative. 2. Derivation From the Limit Definition Start from the definition of the derivative as a limit of a difference quotient: The numerator f(x+h)g(x+h) - f(x)g(x) won't factor as it stands. The key trick is to add and subtract — adding zero, but in a form that lets us group: Each group is a single function times a difference quotient. Splitting the limit and letting — so by continuity — collapses the quotients into g'(x) and f'(x) : 3. Why Two Terms? The Area Picture Here is the most intuitive reason two terms appear. Read f(x) and g(x) as the side lengths of a rectangle, so its area is the product . Now let both sides grow a little, by and . Two long strips — and — plus a tiny corner . Divide by and take the limit as : the corner shrinks quadratically and vanishes, while the two strips survive as exactly the two Product-Rule terms.

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