Product Rule

Core Idea

The product rule states that d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x). The derivative of a product is NOT the product of the derivatives. Instead, you differentiate each factor while keeping the other unchanged, then add the results. This rule is necessary whenever two non-constant functions are multiplied together.

How It's Best Learned

Derive from the limit definition by adding and subtracting f(x+h)*g(x). Practice with products of polynomials (verifiable by expanding first), then with products involving trig and exponential functions. Use the mnemonic "first times derivative of second plus second times derivative of first."

Common Misconceptions

• Believing (fg)' = f'g': this is the single most common calculus error.
• Forgetting one of the two terms in the product rule.
• Not recognizing when the product rule is needed vs. when the constant multiple rule suffices (if one factor is a constant, use the simpler rule).

Explainer

From your work with the constant multiple and sum rules, you know that differentiation is linear: you can pull out constants and differentiate term by term. A natural next question is whether multiplication works similarly — can you just differentiate each factor and multiply the results? The answer is no, and understanding why is the first step to internalizing the product rule. Consider f(x) = x and g(x) = x, so f(x)g(x) = x². If (fg)' = f'g', you would get 1·1 = 1. But d/dx[x²] = 2x. The error grows with x because multiplying two changing quantities creates an interaction that the "differentiate each part separately" idea misses.

The correct formula is d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). A geometric analogy makes this intuitive. Imagine a rectangle with side lengths f(x) and g(x). Its area is A(x) = f(x)g(x). When x increases by a tiny amount Δx, both sides grow: f increases by Δf ≈ f'Δx and g increases by Δg ≈ g'Δx. The new area has three new pieces: a strip of width Δf and height g (contributing f'g Δx), a strip of width f and height Δg (contributing fg' Δx), and a tiny corner of area ΔfΔg (which is negligible in the limit because it is second-order in Δx). So ΔA ≈ (f'g + fg')Δx, giving dA/dx = f'g + fg'. Each term in the product rule corresponds to one strip of the rectangle.

Applying the rule is straightforward once you identify the two factors. For h(x) = x³sin(x), take f = x³ and g = sin(x). Then f' = 3x² and g' = cos(x), giving h'(x) = 3x²·sin(x) + x³·cos(x). The key habit is to always write out both terms — skipping one is the most common error. The mnemonic "first times derivative of second, plus second times derivative of first" (or "d-first, d-second") helps. Note also when the rule is not needed: if one factor is a constant, like h(x) = 5sin(x), the constant multiple rule gives h'(x) = 5cos(x) directly. The product rule is only necessary when both factors depend on x.

The product rule also reveals why integration by parts — which you will encounter next — works the way it does. Rearranging the product rule: f(x)g'(x) = [f(x)g(x)]' − f'(x)g(x). Integrating both sides gives the integration-by-parts formula. So the product rule is not just a differentiation technique; it is the foundation of one of the most important integration strategies in calculus. Internalizing it now pays dividends throughout the rest of the course.