Here is the infinitesimal definition of the derivative that Leibniz (and Newton implicitly) used: df = f(x+dx) - f(x), and df/dx = (f(x+dx) - f(x))/dx. For instance, if f(x) = x^2, then df = (x+dx)^2 - x^2 = 2xdx + dx^2, and then you get rid of dx^2 because it's the square of an infinitesimal, so it's infinitely smaller than 2xdx. Thus we have df = 2xdx, so df/dx = 2x.DeeAytch said:Thank you for responding.
I am looking specifically for examples of the algorithms as they were employed historically. If nobody can provide that, then I suppose I'll dig through the sources you listed.
Newton and Leibniz both independently developed the concept of calculus, including the method of differentiation. However, Newton's approach was based on his theory of fluxions, while Leibniz's approach was based on his theory of infinitesimal calculus. This led to some differences in notation and terminology, but the underlying principles are essentially the same.
Many mathematicians and scientists argue that Leibniz's approach is more intuitive, as it is based on the concept of infinitely small quantities that are easy to visualize and understand. Newton's approach, on the other hand, involves the use of limits and can be more abstract.
The Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations, plays a central role in both Newton and Leibniz's approaches to calculus. It allows for the calculation of areas under curves and the evaluation of antiderivatives, making it a fundamental concept in the study of calculus.
Yes, there was a significant controversy surrounding the development of calculus by Newton and Leibniz. Both claimed to have independently developed the method of differentiation, leading to a heated debate and accusations of plagiarism. The controversy was eventually settled, with both Newton and Leibniz recognized as pioneers of calculus.
The approach to differentiation has evolved significantly since the time of Newton and Leibniz. The concepts of limits, continuity, and differentiability have been formalized, and the notation and terminology have been standardized. Additionally, more advanced techniques and applications of differentiation have been developed, such as the chain rule, implicit differentiation, and differential equations.