f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}Variables
- f(x): original function
- h: change in the input
- f′(x): derivative at x
How to use this formula
Defines the instantaneous rate of change of a function as a limit.
Important notes
- The limit must exist for the derivative to exist.
- The quotient represents the slope of a secant line before h approaches zero.
Quick example
Applying the definition to f(x) = x² gives f′(x) = 2x.
Applicability, worked calculation, and verification
Domain and applicability
The derivative exists at x only when the difference quotient approaches a finite common limit from all required directions.
Units
- derivative units are function-output units per x-unit
Assumptions and domain checks
- The limit must exist for the derivative to exist.
- For the Derivative Definition, 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 Derivative Definition without checking differentiability, integrability, convergence, interval, and transform-convention requirements.
Boundary and special cases
- For Derivative Definition, check zero, negative, and extreme input values before relying on the result.
- When using Derivative Definition, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For f(x) = x², [f(x+h) − f(x)]/h = [(x+h)² − x²]/h = 2x + h. Taking h → 0 gives f′(x) = 2x.
- Substitute f(x) = x² into the difference quotient.
- Expand (x + h)² and subtract x² to obtain 2xh + h².
- Factor h and cancel it for h ≠ 0, leaving 2x + h.
- Take the limit as h approaches 0 to obtain 2x.
Independent verification
The result agrees with the power rule, and numerical secant slopes near a chosen x approach 2x as h becomes smaller.
Common mistakes
- When copying Derivative Definition, 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 Derivative Definition 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
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}. - 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 Derivative Definition.
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 Derivative Definition used for?
Defines the instantaneous rate of change of a function as a limit.
Can I copy this formula as LaTeX?
Yes. Copy f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} 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.