## History and applications

### Mathematically rigorous calculus

Both Newton's and Leibniz's versions of the calculus fell far short of the standards of rigour demanded by mathematicians today. Leibniz's infinitesimals like \(dx\) and Newton's fluxions were concepts that many argued were poorly defined or incoherent.

The most scathing criticism perhaps came from Bishop Berkeley (The Analyst, 1734), who ridiculed fluxions and infinitesimals:

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

It was not until over a century later that ideas like limits were formally introduced, and put on a firm mathematical footing, so that today we present the derivative as

\[ f'(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x}. \]As with many branches of mathematics, the way that calculus is taught and learned bears little relation to its historical development.