Product rule
Differentiate a product without treating it as two separate derivatives.
Product rule is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why product rule works
When two quantities multiply, the product grows for two separate reasons: the first factor changes while the second sits still, and the second changes while the first sits still. Adding those two contributions gives the rule. The piece where both change at once is too small to survive the limit.
What each symbol means
$f,g$ are differentiable functions of the same variable.
Product rule: when it holds
Both functions must be differentiable at the point.
When it stops applying
If one factor has no derivative at the point, the rule cannot be used even when the product is perfectly smooth there. The function x times |x| has derivative 0 at the origin, yet |x| has no derivative there at all, so the answer must be found from the definition.
Product rule: a worked example
$\frac{d}{dx}(x^2e^x)=2xe^x+x^2e^x$.
The mistake to avoid
What people do: Writing the derivative of a product as the product of the two derivatives.
Why it goes wrong: A quick test kills it: with both factors equal to x the shortcut gives 1, while the true derivative of x squared is 2x.
Do this instead: Differentiate one factor at a time, leave the other one untouched, and add the two results together.
Product rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $f,g$ are differentiable functions of the same variable.
- Check the conditions before substituting. Both functions must be differentiable at the point.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where product rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about product rule
Does the order of the two terms matter?
No, because both multiplication and addition can be rearranged freely here. Picking one order and sticking to it is still worth doing, since it is how you notice a missing term.
Can I multiply the two functions out first instead?
Yes when the product expands easily, such as x squared times x cubed. For something like x squared times e to the x there is nothing to expand, so the rule is the only route.
How does this work with three factors multiplied together?
Differentiate one at a time and add three terms. Each term differentiates exactly one of the factors and copies the other two down unchanged.
Why is there no subtraction here like there is for a quotient?
Both factors push the product upward when they grow, so their effects add. Division reverses what the bottom function does, and that reversal is where subtraction comes from.