![]() This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. In mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. The process of finding the derivative is called differentiation. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane.ĭifferential calculus is the study of the definition, properties, and applications of the derivative of a function. ![]() Bernhard Riemann used these ideas to give a precise definition of the integral. Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". In his work Weierstrass formalized the concept of limit and eliminated infinitesimals. In Cauchy's Cours d'Analyse, we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. The foundations of differential and integral calculus had been laid. Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. ![]() 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. ![]() From the age of Greek mathematics, Eudoxus (c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. Which represents the slope of the tangent line at the point (−1,−32).The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken.Įxample 1: Find f′( x) if f( x) = (3x 2 + 5x − 2) 8.Įxample 2: Find f′( x) if f( x) = tan (sec x).Įxample 5: Find the slope of the tangent line to a curve y = ( x 2 − 3) 5 at the point (−1, −32).īecause the slope of the tangent line to a curve is the derivative, you find that Here, three functions- m, n, and p-make up the composition function r hence, you have to consider the derivatives m′, n′, and p′ in differentiating r( x). If a composite function r( x) is defined as Note that because two functions, g and h, make up the composite function f, you have to consider the derivatives g′ and h′ in differentiating f( x). For example, if a composite function f( x) is defined as The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. ![]() Volumes of Solids with Known Cross Sections.Second Derivative Test for Local Extrema.First Derivative Test for Local Extrema.Differentiation of Exponential and Logarithmic Functions.Differentiation of Inverse Trigonometric Functions.Limits Involving Trigonometric Functions. ![]()
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