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In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
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.
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today. Since the 17th century many mathematicians have contributed to the theory of differentiation.
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
Leibniz developed much of the notation used in calculus today. [31]: 51–52 The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.
The general Leibniz rule, [45] named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if f {\displaystyle f} and g {\displaystyle g} are n {\displaystyle n} -times differentiable functions , then the product f g {\displaystyle fg} is also n {\displaystyle n} -times ...
Leibniz's notation for differentiation does not require assigning meaning to symbols such as dx or dy (known as differentials) on their own, and some authors do not attempt to assign these symbols meaning. [1] Leibniz treated these symbols as infinitesimals. Later authors have assigned them other meanings, such as infinitesimals in non-standard ...