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With hindsight, however, it is considered the first general theorem of calculus to be discovered. [1] The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse ...
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 ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in: Mathematical Handbook of Formulas and Tables (3rd edition) , S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISBN 978-0-07-154855-7 .
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V .
The study of differential calculus is unified with the calculus of finite differences in time scale calculus. [54] The arithmetic derivative involves the function that is defined for the integers by the prime factorization. This is an analogy with the product rule. [55]
In geometric calculus, the geometric derivative satisfies a weaker form of the Leibniz (product) rule. It specializes the Fréchet derivative to the objects of geometric algebra. Geometric calculus is a powerful formalism that has been shown to encompass the similar frameworks of differential forms and differential geometry. [1]
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