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Derivative Is A Complex Subject Of Calculus

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Derivative is a complex subject of calculus. In calculus, derivative is a key term developed by both Newton and Leibniz. With function f (t), the first derivative is defined asf^ ' (t)= df/dt= lim┬(h→0)⁡〖(f(t)-f(t-h))/h〗. There is also a second derivative known as second-order derivative. The second-order derivative is defined: f^ ' ' (t)= (d^2 f)/(dt^2 )= lim┬(h→0)⁡〖(f^ ' (t)-f^ ' (t-h))/h〗
=lim┬(h→0)⁡〖1/h {(f(t)-f(t-h))/h- (f(t-h)-f(t-2h))/h}〗
=lim┬(h→0)⁡〖(f(t)-2f(t-h)+f(t-2h))/h^2 〗
(Podlubny, 1998). The general idea of derivative has been for several times as the topic has been dropped and brought up again for further research. Over the years, derivative has been defined in many algorithms and methods, such as methods for the…show more content…
However, early mathematicians solved these questions with what later became known as derivative. Derivative has developed or created the ideas of extrema, tangent, and limit (Anderson, Katz, and Wilson, 2004). Derivative contributed towards the mathematical world from an early stage to the development of other rules. Derivative was mentioned early in the 1660’s to develop new methods and rules. Many of the methods known today and that have been developed over the years all derived because of the use of derivative. Derivative supports the evidence of many proofs such as extreme, tangent, and limit. Newton and Leibniz both saw derivative in a different way. Newton and Leibniz both created calculus separately, as in Newton and Leibniz were not working together for the creation of calculus. They both created or invented calculus by the development of derivative. For example, in the first step of creating calculus, Newton and Leibniz took the idea and concepts of derivative and integral. The general concepts of derivative and integral came from the methods for finding tangents, extrema, and area. The general concept of derivative and integral came from the methods of finding other solutions. Another invention they came across while inventing calculus was the notation to solve the concepts of derivative and
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