The Product Rule

Calculus 1 · Axiom Academy

Differentiating a product of two functions — derived from the limit, and seen as a growing rectangle. The tempting move is to differentiate each factor and multiply the results. The animation shows why that fails and what replaces it: for two differentiable functions f(x) and g(x) , the derivative of their product is a sum of two terms , each carrying one derivative. The wrong guess — multiplying derivatives The Product Rule — the correct statement 2. Why It Works: A Growing Rectangle Picture a rectangle that is f wide and g tall, so its area is the product . Now nudge x a little: the width grows by and the height by . Watch how the extra area breaks into pieces. The change in area is exactly: Divide by and let . The corner term carries a second factor that also goes to zero, so it disappears — leaving precisely f'g + fg' , the Product Rule. 3. Applying It to a Real Product Take . Set f(x) = x^2 (the width) and (the height). Their derivatives are f'(x) = 2x and . The animation builds the answer the same way the rectangle did — one strip per term. The rectangle gives the intuition; the limit makes it rigorous. Start from the definition of the derivative of the product: The key trick is to add and subtract in the numerator — it changes nothing, but it lets the numerator split into two groups. The animation shows that split converging. Group and factor, then split the limit:

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