Product Rule or Quotient Rule?
Choose the derivative rule from the top-level operation—and simplify first when algebra can remove the choice entirely.
LaTeX article Updated July 11, 2026
Read the top-level operation
Parentheses matter. In x² sin x, multiplication is the final operation, so use the Product Rule. In sin(x²), sine is the final operation, so use the Chain Rule instead.
A quotient can be rewritten as multiplication by a negative power, but that is only helpful when the rewritten structure is simpler.
Simplify before differentiating
Cancel common factors, split simple fractions, and combine powers before committing to a rule. A shorter equivalent expression usually produces a shorter derivative and fewer sign errors.
Do not cancel terms across addition; simplification must remain algebraically valid.
Why both terms are necessary
In a product, each factor changes while the other provides scale. The derivative includes one contribution from f changing and another from g changing. Keeping only f′g misses half the motion.
Worked example
Common mistakes
- Using f′g′ for the derivative of a product.
- Reversing the numerator order in the Quotient Rule.
- Using a large rule before checking for a simple algebraic rewrite.
Keep these ideas
- Identify the final operation.
- Simplify first whenever possible.
- Products need two derivative contributions.