\frac{d}{dx}f(g(x))=f'(g(x))g'(x)Variables
- f: outer function
- g: inner function
- x: independent variable
How to use this formula
Differentiates a composite function by combining outer and inner derivatives.
Important notes
- Differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative.
- The rule can be applied repeatedly to nested compositions.
Quick example
The derivative of sin(x²) is 2x cos(x²).
Applicability, worked calculation, and verification
Domain and applicability
Applies when a differentiable outer function is composed with a differentiable inner function at the point being evaluated.
Units
- derivative units are output units per input unit
Assumptions and domain checks
- Differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative.
- For the Chain Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- 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 Chain Rule without checking differentiability, integrability, convergence, interval, and transform-convention requirements.
Boundary and special cases
- For Chain Rule, check zero, negative, and extreme input values before relying on the result.
- When using Chain Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \frac{d}{dx}f(g(x))=f'(g(x))g'(x) for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For y = (3x + 1)⁴, let u = 3x + 1. Then dy/dx = 4u³·3 = 12(3x + 1)³.
- Identify the outer function u⁴ and the inner function u = 3x + 1.
- Differentiate the outer function with respect to u: 4u³.
- Differentiate the inner function with respect to x: du/dx = 3.
- Multiply the derivatives and replace u to obtain 12(3x + 1)³.
Independent verification
Expanding (3x + 1)⁴ first and differentiating term by term produces the same polynomial as expanding 12(3x + 1)³.
Common mistakes
- When copying Chain Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks.
Continue the workflow
Use Chain 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
\frac{d}{dx}f(g(x))=f'(g(x))g'(x). - 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 Chain 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 Chain Rule used for?
Differentiates a composite function by combining outer and inner derivatives.
Can I copy this formula as LaTeX?
Yes. Copy \frac{d}{dx}f(g(x))=f'(g(x))g'(x) 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.