What Is the Product Rule?
The product rule tells you how to differentiate a function that's the product of two other functions, both of which depend on x. If f(x) = u(x)·v(x), then:
f'(x) = u'(x)·v(x) + u(x)·v'(x)
In words: differentiate the first factor and leave the second alone, then leave the first alone and differentiate the second, and add the two results. The rule exists because differentiation does not distribute over multiplication the way it does over addition — the derivative of a product is emphatically not the product of the derivatives, a fact you can verify with the simplest example: d/dx[x·x] = d/dx[x²] = 2x, while the product of the individual derivatives would be 1·1 = 1.
How to Use This Calculator
Enter a product of two functions, like x^2*sin(x) or x*exp(x). The calculator splits the expression into its two factors u and v, differentiates each independently, writes out the u'v + uv' combination explicitly, and then simplifies the final result. Each stage is displayed as a separate step so you can compare against your own working line by line — the most useful part for finding exactly where a mistake crept in. An optional evaluation point gives you the numeric slope, and the graph shows both the product function and its derivative.
The Product Rule at a Glance
| Function | u | v | Derivative u'v + uv' |
|---|---|---|---|
| x²·sin(x) | x² | sin(x) | 2x·sin(x) + x²·cos(x) |
| x·eˣ | x | eˣ | eˣ + x·eˣ |
| eˣ·cos(x) | eˣ | cos(x) | eˣcos(x) − eˣsin(x) |
| x³·ln(x) | x³ | ln(x) | 3x²ln(x) + x² |
Worked Example
Differentiate f(x) = x²·sin(x). Assign u = x² and v = sin(x); then u' = 2x and v' = cos(x). Substituting into u'v + uv':
f'(x) = (2x)(sin x) + (x²)(cos x) = 2x·sin(x) + x²·cos(x)
Both terms survive in the final answer — there's usually no cancellation in a product rule result, which is itself a useful sanity check: if your product rule answer collapsed to a single term, double-check whether you dropped one half of the formula.
A second check worth building into your habits: evaluate both your answer and a finite-difference estimate at one test point. Here, f'(1) = 2sin(1) + cos(1) ≈ 2.2232, and the numeric slope [f(1.001) − f(0.999)]/0.002 gives the same value to four decimals — exactly the verification this calculator performs when you supply an evaluation point.
Common Mistakes to Avoid
- Multiplying the derivatives together. d/dx[uv] ≠ u'·v'. The correct formula keeps one factor undifferentiated in each term.
- Dropping one of the two terms. Under time pressure, it's common to write only u'v and forget uv' (or vice versa). The rule always produces a sum of two terms.
- Sign confusion with the quotient rule. The product rule uses a plus between its terms; the quotient rule uses a minus and a squared denominator. Mixing the two formulas up is a classic exam error — see the Quotient Rule Calculator for the contrast.
- Missing a chain rule inside one factor. For x²·sin(3x), differentiating sin(3x) requires the chain rule (giving 3cos(3x)), nested inside the product rule. Rules combine constantly in real problems.
Real-World Applications of the Product Rule
Quantities that are products of two changing things appear everywhere, and the product rule is how their rates of change are analyzed. Revenue is price times quantity sold — and when both price and demand change over time (or price changes affect demand), the rate of change of revenue is a textbook product rule computation: R'(t) = p'(t)q(t) + p(t)q'(t). This decomposition is genuinely informative: it splits revenue growth into the part driven by price changes and the part driven by volume changes, which is exactly how analysts decompose business performance.
In physics, momentum is mass times velocity, and for systems whose mass changes over time — a rocket burning fuel is the canonical example — Newton's second law must be applied in the form F = d(mv)/dt, which expands by the product rule into m'v + mv'. That extra m'v term (thrust from expelled mass) is the entire principle behind rocket propulsion. In biology and pharmacology, the amount of a substance in a system often equals concentration times volume, both varying with time, and tracking the total's rate of change again requires the product rule.
Frequently Asked Questions
Does it matter which factor I call u and which I call v?
No — the formula u'v + uv' is symmetric, so swapping the labels gives the same final answer with the two terms in the opposite order. Pick whichever assignment feels natural.
What if there are three factors multiplied together?
The rule extends naturally: d/dx[uvw] = u'vw + uv'w + uvw' — differentiate each factor once while leaving the others alone, and sum the results. You can also apply the two-factor rule twice by grouping: (uv)·w. This calculator handles multi-factor products by applying the rule recursively.
Why isn't the derivative of a product just the product of the derivatives?
Because both factors are changing simultaneously, and each factor's change gets scaled by the current size of the other factor. The u'v term captures how u's change matters in proportion to v's current value, and vice versa — multiplying the derivatives together would ignore the sizes of the functions entirely.
When should I use the product rule versus just expanding first?
For polynomial products like x²(x+1), expanding to x³+x² and using the power rule is often faster. The product rule becomes essential when the factors can't be multiplied out — like x²·sin(x) or eˣ·ln(x) — which is most of the time in practice.
Is there a product rule for integration too?
The closest counterpart is integration by parts, which is derived directly by integrating the product rule formula and rearranging. It's the standard technique for integrals like ∫x·eˣ dx.
How does the product rule generalize to higher derivatives?
The second derivative of a product expands to (uv)'' = u''v + 2u'v' + uv'', and in general the nth derivative follows Leibniz's rule: a sum of terms with binomial coefficients, (uv)⁽ⁿ⁾ = Σ C(n,k)·u⁽ᵏ⁾·v⁽ⁿ⁻ᵏ⁾ — structurally identical to the binomial theorem for (a+b)ⁿ. That parallel isn't a coincidence: both formulas count the ways of distributing n operations between two objects. For repeated differentiation in practice, the Higher-Order Derivative Calculator applies the expansion automatically at each order.