Math 2.0

Real functions Definition

Difference quotient

Builds on FunctionDerivativeLimit of a functionMean value theorem

Definition

Let D⊆ℝD \subseteq \R and let f ⁣:D→ℝf \colon D \to \R be a function. Let x∈Dx \in D and let h≠0h \ne 0 be a real number with x+h∈Dx + h \in D. The difference quotient of ff at xx with increment hh is f(x+h)−f(x)h.\frac{f(x + h) - f(x)}{h}. Equivalently, for a≠ba \ne b in DD, the difference quotient of ff between aa and bb is f(b)−f(a)b−a\dfrac{f(b) - f(a)}{b - a}.

Remarks

The function ff has derivative f′(x)f'(x) at an interior point xx of DD exactly when the difference quotient has the limit f′(x)f'(x) as h→0h \to 0. The mean value theorem says that, for ff continuous on [a,b][a, b] and differentiable on (a,b)(a, b), the difference quotient between aa and bb equals f′(c)f'(c) for some c∈(a,b)c \in (a, b).

Example: for f(x)=x2f(x) = x^2, the difference quotient is (x+h)2−x2h=2xh+h2h=2x+h\frac{(x + h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h. It tends to 2x2x as h→0h \to 0, which is f′(x)f'(x).

Non-example: the increment h=0h = 0 is not allowed, since the expression f(x)−f(x)0\frac{f(x) - f(x)}{0} has no meaning. This is why the derivative is a limit and not a value of the quotient.

In numerical analysis, the quotient is also called the forward difference quotient, and f(x+h)−f(x−h)2h\frac{f(x + h) - f(x - h)}{2h} is the central difference quotient. The symbol ΔyΔx\frac{\Delta y}{\Delta x} is common notation for it.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionProposition, concept, no evidence levelPropositionconceptSequence, definition, no evidence levelSequencedefinitionSet, definition, no evidence levelSetdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionCartesian product, definition, no evidence levelCartesian productdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionSubset, definition, no evidence levelSubsetdefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBinary relation, definition, no evidence levelBinary relationdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementFunction, definition, no evidence levelFunctiondefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementDifference quotient, definition, no evidence levelDifference quotientdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Limit point of a set definition
  3. Local maximum definition
  4. Proposition concept
  5. Sequence definition
  6. Set definition
  7. Limit of a function definition
  8. Local minimum definition
  9. Implication definition
  10. Predicate definition
  11. Real sequence definition
  12. Subsequence definition
  13. Ordered pair definition
  14. Continuity at a point definition
  15. Derivative definition
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Bounded sequence definition
  19. Limit of a sequence definition
  20. Monotone sequence definition
  21. Cartesian product definition
  22. Continuous function definition
  23. Subset definition
  24. Fermat's theorem on stationary points statement
  25. Uniqueness quantifier definition
  26. Convergent sequence definition
  27. Monotone convergence theorem for sequences statement
  28. Every sequence has a monotone subsequence statement
  29. Binary relation definition
  30. Bolzano–Weierstrass theorem statement
  31. Boundedness theorem statement
  32. Function definition
  33. Extreme value theorem statement
  34. Rolle's theorem statement
  35. Mean value theorem statement
  36. Difference quotient definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Difference_quotient
About this page
Tag97DDM0ZBYG
URIm2://97DDM0ZBYG
Kinddefinition
WritingW0 stub
MSC26A24, 26A06
DefinesDifference quotient
UsesFunction, Derivative, Limit of a function, Mean value theorem
Changed2 November 2026

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