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  2. Differentiable function - Wikipedia

    en.wikipedia.org/wiki/Differentiable_function

    If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists. In other words, the graph of f has a non-vertical tangent line at the point (x 0, f(x 0)). f is said to be differentiable on U if it is differentiable at every point of U.

  3. Differential of a function - Wikipedia

    en.wikipedia.org/wiki/Differential_of_a_function

    In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).

  4. C0-semigroup - Wikipedia

    en.wikipedia.org/wiki/C0-semigroup

    An equivalent characterization in terms of Cauchy problems is the following: the strongly continuous semigroup generated by A is eventually differentiable if and only if there exists a t 1 ≥ 0 such that for all xX the solution u of the abstract Cauchy problem is differentiable on (t 1, ∞).

  5. Inverse function theorem - Wikipedia

    en.wikipedia.org/wiki/Inverse_function_theorem

    The inverse function theorem can also be generalized to differentiable maps between Banach spaces X and Y. [20] Let U be an open neighbourhood of the origin in X and : a continuously differentiable function, and assume that the Fréchet derivative : of F at 0 is a bounded linear isomorphism of X onto Y.

  6. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    The mean value theorem gives a relationship between values of the derivative and values of the original function. If f(x) is a real-valued function and a and b are numbers with a < b, then the mean value theorem says that under mild hypotheses, the slope between the two points (a, f(a)) and (b, f(b)) is equal to the slope of the tangent line to ...

  7. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing ⁠ ⁠, and the limit = (+) exists. [2] This means that, for every positive real number ⁠ ⁠, there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.

  8. Chain rule - Wikipedia

    en.wikipedia.org/wiki/Chain_rule

    In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.

  9. Leibniz's notation - Wikipedia

    en.wikipedia.org/wiki/Leibniz's_notation

    where x is thought of as a function of a new variable u and the function y on the left is expressed in terms of x while on the right it is expressed in terms of u. If y = f(x) where f is a differentiable function that is invertible, the derivative of the inverse function, if it exists, can be given by, [21]