KnowraDifferentiabilityLinked fromLinked fromThe 25 pages that link to Differentiability, each with the reason it gives.All 25Broader topic 1Related 14Narrower topic 8Compared with 2DerivativeRelated: It distinguishes points where the instantaneous rate is well-defined from points where it is not.ContinuityCompared with: Differentiability implies continuity, but continuous functions need not be differentiable.Chain ruleRelated: The standard chain rule requires differentiability of the component functions at the relevant points.Real analysisRelated: Derivatives arise from limits of difference quotients.Tangent lineRelated: Differentiability guarantees a well-defined tangent direction for a graph at that point.Taylor's theoremRelated: The theorem requires enough derivatives for the stated polynomial degree and remainder.Lipschitz continuityCompared with: Lipschitz functions need not be differentiable at every point.Karl WeierstrassRelated: The Weierstrass function shows that continuity alone does not guarantee differentiability anywhere.AntiderivativeNarrower topic: Being a derivative is a stronger condition than merely being continuous.Mean value theoremNarrower topic: Differentiability at every interior point is the theorem’s other essential hypothesis.Weierstrass functionNarrower topic: The example separates continuity from the stronger requirement of a local linear approximation.Rolle's theoremNarrower topic: Differentiability inside the interval makes Fermat's stationary-point result applicable.Darboux's theoremNarrower topic: The theorem applies to derivatives produced by differentiable functions.Implicit differentiationRelated: A smooth local curve is needed for its slope to be described by a derivative.L'Hôpital's ruleRelated: The numerator and denominator must be differentiable in the relevant punctured interval.Linear approximationNarrower topic: This condition determines when the approximation is genuinely accurate to first order.Envelope theoremRelated: Differentiability conditions determine when the envelope formula yields an ordinary derivative.Implicit functionNarrower topic: The standard implicit function theorem requires differentiability of the defining relation.Cauchy's mean value theoremNarrower topic: Interior differentiability supplies the derivative values compared by the theorem.Hotelling's lemmaRelated: The standard derivative form of the lemma requires a differentiable profit function.Roy's identityRelated: The ratio requires the indirect utility function to have usable price and income derivatives.Rademacher's theoremBroader topic: The theorem guarantees this local linear approximation outside a measure-zero set.Differentiation rulesRelated: Rules require differentiable inputs wherever their formulas are applied.Interior extremum theoremRelated: Differentiability supplies the local rate of change whose value the theorem constrains.Symmetry of second derivativesRelated: Existence of second partial derivatives alone does not ensure their symmetry.
KnowraDifferentiabilityLinked fromLinked fromThe 25 pages that link to Differentiability, each with the reason it gives.All 25Broader topic 1Related 14Narrower topic 8Compared with 2DerivativeRelated: It distinguishes points where the instantaneous rate is well-defined from points where it is not.ContinuityCompared with: Differentiability implies continuity, but continuous functions need not be differentiable.Chain ruleRelated: The standard chain rule requires differentiability of the component functions at the relevant points.Real analysisRelated: Derivatives arise from limits of difference quotients.Tangent lineRelated: Differentiability guarantees a well-defined tangent direction for a graph at that point.Taylor's theoremRelated: The theorem requires enough derivatives for the stated polynomial degree and remainder.Lipschitz continuityCompared with: Lipschitz functions need not be differentiable at every point.Karl WeierstrassRelated: The Weierstrass function shows that continuity alone does not guarantee differentiability anywhere.AntiderivativeNarrower topic: Being a derivative is a stronger condition than merely being continuous.Mean value theoremNarrower topic: Differentiability at every interior point is the theorem’s other essential hypothesis.Weierstrass functionNarrower topic: The example separates continuity from the stronger requirement of a local linear approximation.Rolle's theoremNarrower topic: Differentiability inside the interval makes Fermat's stationary-point result applicable.Darboux's theoremNarrower topic: The theorem applies to derivatives produced by differentiable functions.Implicit differentiationRelated: A smooth local curve is needed for its slope to be described by a derivative.L'Hôpital's ruleRelated: The numerator and denominator must be differentiable in the relevant punctured interval.Linear approximationNarrower topic: This condition determines when the approximation is genuinely accurate to first order.Envelope theoremRelated: Differentiability conditions determine when the envelope formula yields an ordinary derivative.Implicit functionNarrower topic: The standard implicit function theorem requires differentiability of the defining relation.Cauchy's mean value theoremNarrower topic: Interior differentiability supplies the derivative values compared by the theorem.Hotelling's lemmaRelated: The standard derivative form of the lemma requires a differentiable profit function.Roy's identityRelated: The ratio requires the indirect utility function to have usable price and income derivatives.Rademacher's theoremBroader topic: The theorem guarantees this local linear approximation outside a measure-zero set.Differentiation rulesRelated: Rules require differentiable inputs wherever their formulas are applied.Interior extremum theoremRelated: Differentiability supplies the local rate of change whose value the theorem constrains.Symmetry of second derivativesRelated: Existence of second partial derivatives alone does not ensure their symmetry.