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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 ...
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.
A function is differentiable at an interior point a of its domain if and only if it is semi-differentiable at a and the left derivative is equal to the right derivative. An example of a semi-differentiable function, which is not differentiable, is the absolute value function () = | |, at a = 0.
If there exists an m × n matrix A such that = + ‖ ‖ in which the vector ε → 0 as Δx → 0, then f is by definition differentiable at the point x. The matrix A is sometimes known as the Jacobian matrix , and the linear transformation that associates to the increment Δ x ∈ R n the vector A Δ x ∈ R m is, in this general setting ...
Differentiable function – Mathematical function whose derivative exists Differential of a function – Notion in calculus Differentiation of integrals – Problem in mathematics
It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable. If f is differentiable at a point x 0, then f must also be continuous at x 0. In particular, any differentiable function must ...
If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique. Also, if u is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative ...
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one.