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  2. Mean value theorem (divided differences) - Wikipedia

    en.wikipedia.org/wiki/Mean_value_theorem...

    For any n + 1 pairwise distinct points x ... , x n in the domain of an n-times differentiable ... where the nth derivative of f equals n ! times the nth divided ...

  3. Strict differentiability - Wikipedia

    en.wikipedia.org/wiki/Strict_differentiability

    Let K be a complete extension of Q p (for example K = C p), and let X be a subset of K with no isolated points. Then a function F : X → K is said to be strictly differentiable at x = a if the limit (,) (,) () exists.

  4. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    The absolute value function is continuous but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do from the right. If f {\displaystyle f} is differentiable at ⁠ a {\displaystyle a} ⁠ , then f {\displaystyle f} must also be continuous at a {\displaystyle a} . [ 11 ]

  5. Time-scale calculus - Wikipedia

    en.wikipedia.org/wiki/Time-scale_calculus

    In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems.

  6. Differential (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Differential_(mathematics)

    For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically ...

  7. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.

  8. Smoothness - Wikipedia

    en.wikipedia.org/wiki/Smoothness

    The C 0 function f (x) = x for x ≥ 0 and 0 otherwise. The function g (x) = x 2 sin(1/ x) for x > 0. The function : with () = ⁡ for and () = is differentiable. However, this function is not continuously differentiable.

  9. Divided differences - Wikipedia

    en.wikipedia.org/wiki/Divided_differences

    In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. [citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. [1] Divided differences is a recursive division process.