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In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), returns a function or value as its result. All other functions are first-order functions.
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
The classical Pade scheme for the first derivative at a cell with index (′) reads; ′ + ′ + + ′ = +. Where is the spacing between points with index , & +.The equation yields a fourth-order accurate solution for ′ when supplemented with suitable boundary conditions (typically periodic).
Derivative test. In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is ...
The derivatives and integrals of calculus can be packaged into the modern theory of differential forms, in which the derivative is genuinely a ratio of two differentials, and the integral likewise behaves in exact accordance with Leibniz notation. However, this requires that derivative and integral first be defined by other means, and as such ...
The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after ...
In calculus, the product rule (or Leibniz rule[1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as or in Leibniz's notation as. The rule may be extended or generalized to products of three or more functions, to a rule for higher ...