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Calculus. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. [ 1][ 2][ 3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is. It is provable in many ways by using other derivative rules .
One proof of the chain rule begins by defining the derivative of the composite function f ∘ g, where we take the limit of the difference quotient for f ∘ g as x approaches a: ′ = (()) (()). Assume for the moment that g ( x ) {\displaystyle g(x)\!} does not equal g ( a ) {\displaystyle g(a)} for any x {\displaystyle x} near a ...
The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives.
Answer: In words: the holomorphic function can be obtained by putting and in . Example 1: with and we obtain . Example 2: with and we obtain . Proof : From the first pair of definitions and . This is an identity even when and are not real, i.e. the two variables and may be considered independent. Putting we get .
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 ...
Danskin's theorem. In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form. The theorem has applications in optimization, where it sometimes is used to solve minimax problems. The original theorem given by J. M. Danskin in his 1967 monograph [ 1] provides a formula for the ...
The list of unsuccessful proposed proofs started with Euler's, published in 1740, [3] although already in 1721 Bernoulli had implicitly assumed the result with no formal justification. [4] Clairaut also published a proposed proof in 1740, with no other attempts until the end of the 18th century. Starting then, for a period of 70 years, a number ...
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real -valued function. The most basic version starts with a real-valued function f, its derivative f ′, and ...