Newton and Leibniz approach to differentiation

[DeeAytch]

Newton and Leibniz both had a method of differentiating. Newton had fluxions and Leibniz had something that resembles the modern derivative.

Historically, does anyone know how they went about calculating the derivative?

 

[Simon Bridge]

This is the sort of stuff you can find online.

Newton:
http://www.gutenberg.org/ebooks/author/6288

Leibnitz
http://www.gutenberg.org/ebooks/search/?query=leibnitz (in German though)

Also:
http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html

... and:
1684. Nova methodus pro maximis et minimis (New method for maximums and minimums); translated in Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard University Press: 271–81.

... should get you started.

 

[DeeAytch]

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.

 

[Simon Bridge]

Both authors had long careers in which they employed their respective techniques via a range of actual processes or algorithms which they developed as they went. The references provide access to a range of examples - you'll see what I mean quite quickly. Good luck.

 

[lugita15]

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.

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.