Product Rule: Step-by-Step Guide
The product rule differentiates a product of two functions: if y = u(x)*v(x), then y' = u'v + u v'. Use it whenever neither factor is a constant and the function is genuinely a product, not a composition.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Identify the two factors. Write the function as a product u(x)*v(x). Choose u and v so that each is straightforward to differentiate.\[y = u(x)\,v(x)\]
- Differentiate each factor separately. Compute u'(x) and v'(x) using whatever rules apply (power, chain, exponential, etc.). Keep them clearly labeled.\[u'(x), \;\; v'(x)\]
- Apply the product rule formula. Plug into y' = u'v + u v'. Note the symmetry: each piece is one factor differentiated times the other untouched.\[\frac{dy}{dx} = u'(x)\,v(x) + u(x)\,v'(x)\]
- Simplify the expression. Distribute, collect like terms, factor out common pieces, and write the answer in its cleanest form.\[\frac{dy}{dx} = \text{simplified}\]
Worked example
Find d/dx of y = x^2 * sin(x).
Step 1: Identify u and v. Let u = x^2 and v = sin(x).
Step 2: Differentiate each factor: u' = 2x and v' = cos(x).
Step 3: Apply the product rule: y' = u'v + u v' = 2x * sin(x) + x^2 * cos(x).
Step 4: Factor x: y' = x(2 sin(x) + x cos(x)).
Step 5: This is fully simplified.
Verification: At x = 0, y' = 0 * (0 + 0) = 0. Numerical: (0.001^2 sin(0.001) - 0)/0.001 approximately 10^-9, confirming derivative near 0.
Tips and common mistakes
- Use the product rule only when neither factor is a constant — the constant multiple rule handles c*f(x).
- The product rule does NOT generalize as (uv)' = u'v' — that is a common mistake.
- For three factors u v w, the rule extends to u'vw + uv'w + uvw'.
- Combine product and chain rules when one factor is itself a composition.
- Sometimes simplifying the product first (e.g., x^2 * x^3 = x^5) avoids the rule entirely.
- Keep track of u, u', v, v' clearly — most product-rule errors come from mixing them up.
- After applying the rule, check whether common factors can be pulled out for a cleaner form.
- When checking your answer, plug in a specific x and compare to a numerical derivative estimate.