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A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. 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 f ′ ( x 0 ) {\displaystyle f'(x_{0})} exists.
Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
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.
Constantin Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule. [6] Under this definition, a function f is differentiable at a point a if and only if there is a function q, continuous at a and such that f(x) − f(a) = q(x)(x − a).
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation . Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is ...
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 ).
To be a C r-loop, the function γ must be r-times continuously differentiable and satisfy γ (k) (a) = γ (k) (b) for 0 ≤ k ≤ r. The parametric curve is simple if | (,): (,) is injective. It is analytic if each component function of γ is an analytic function, that is, it is of class C ω.