(fg)'=f'g+fg'Variables
- f, g: differentiable functions
- f′, g′: their derivatives
How to use this formula
Differentiates the product of two differentiable functions.
Important notes
- Differentiate one factor at a time while keeping the other factor unchanged.
- Do not multiply the two derivatives together.
Quick example
For x²sin x, the derivative is 2x sin x + x² cos x.
Applicability, worked calculation, and verification
Domain and applicability
Applies where both factor functions are differentiable; it is not equivalent to multiplying the two derivatives.
Units
- derivative units are product-output units per input unit
Assumptions and domain checks
- Differentiate one factor at a time while keeping the other factor unchanged.
- The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used.
Do not use this formula when
- Do not use Product Rule without checking differentiability, integrability, convergence, interval, and transform-convention requirements.
Boundary and special cases
- For Product Rule, check zero, negative, and extreme input values before relying on the result.
- When using Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form (fg)'=f'g+fg' for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For f(x) = x² sin x, f′(x) = (2x)sin x + x² cos x.
- Separate the product into u(x) = x² and v(x) = sin x.
- Differentiate each factor: u′(x) = 2x and v′(x) = cos x.
- Apply (uv)′ = u′v + uv′ without dropping either term.
- Substitute the factors to obtain 2x sin x + x² cos x.
Independent verification
At x = 0, both the symbolic derivative and a local numerical slope are 0, providing a simple consistency check.
Common mistakes
- Before substituting values into Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks.
Continue the workflow
Use Product Rule in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
(fg)'=f'g+fg'. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Product Rule.
Verified against: OpenStax Calculus Volume 1
Automated quality check: Passed the core formula indexing gate.
Formula references
- Calculus Volume 1OpenStax, Rice University — Reviewed limits, derivatives, integrals, and foundational calculus formulas.
Frequently asked questions
What is the Product Rule used for?
Differentiates the product of two differentiable functions.
Can I copy this formula as LaTeX?
Yes. Copy (fg)'=f'g+fg' or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.