Cauchy's mean value theorem

For continuous functions on a closed interval and differentiable functions inside it, some interior point makes their difference ratio equal their derivative ratio, when the denominator is nonzero.

Rolle's theorem: A differentiable function equal at a closed interval’s endpoints has a point inside where its derivative is zero. Applying it to a tailored difference of the two functions yields the theorem’s point.
Continuity: A function is continuous when sufficiently small changes in input produce small changes in output. Both functions must be continuous through the interval, including its endpoints.
L'Hôpital's rule: A method for evaluating certain indeterminate limits of function quotients using derivatives. Cauchy's theorem supplies the key intermediate-point relation behind its generalized form.
Taylor's theorem with Cauchy remainder: A form of Taylor’s theorem expressing its remainder using derivatives at an intermediate point. Its proof uses a related two-function mean-value argument.
Auxiliary function: A constructed function used to transform a problem into a form with known properties. The proof builds one whose endpoint values agree, exposing a zero derivative.
Differentiability: A function is differentiable at a point when its derivative exists there. Interior differentiability supplies the derivative values compared by the theorem.
Taylor's theorem: A theorem approximating a sufficiently differentiable function near a point by a polynomial and a remainder. Cauchy’s theorem is used in proofs that establish forms of Taylor’s remainder.
Generalized mean value theorem: A broad label for results relating changes in multiple functions to derivatives at an intermediate point. Cauchy’s theorem is a standard instance of this broader family.
Lagrange's mean value theorem: A theorem equating a function’s average rate of change on an interval with its derivative at an interior point. It follows by taking one function to be the identity function.
Derivative: A derivative is the limit of a function’s rate of change as the input increment approaches zero. The theorem links a ratio of derivatives to a ratio of finite differences.

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Guillaume de l’HôpitalNarrower topic: Modern proofs of l’Hôpital’s rule commonly derive it from this theorem.