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A function that has infinitely many derivatives is called infinitely differentiable or smooth. [33] Any polynomial function is infinitely differentiable; taking derivatives repeatedly will eventually result in a constant function, and all subsequent derivatives of that function are zero. [34] One application of higher-order derivatives is in ...
This states that differentiation is the reverse process to integration. Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified ...
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.
Differentiation with respect to time or one of the other variables requires application of the chain rule, [1] since most problems involve several variables. Fundamentally, if a function F {\displaystyle F} is defined such that F = f ( x ) {\displaystyle F=f(x)} , then the derivative of the function F {\displaystyle F} can be taken with respect ...
For an advanced application of this analysis to topology of manifolds, see Morse theory. In addition to n th derivatives for any natural number n, there are various ways to define derivatives of fractional or negative orders, which are studied in fractional calculus.
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.