The indispensable derivative
At first glance, the derivative, at the frontier between physics and mathematics, is nothing other than the instantaneous velocity of a moving object. But how to define it more precisely? Try then, without using limits... we seem to divide zero by itself! The greatest minds, from Fermat to Cauchy, passing through Newton, took centuries to rigorously and completely generalize this concept. Yet, it is found everywhere, in everything that varies; it allows us to predict the future in any deterministic phenomenon! A current example: your tax rate and the derivative get along very well...
All articles in this folder
From intuition to rigor
Studying derivatives of functions sometimes leaves only a distant memory of applying somewhat esoteric formulas. Yet the underlying idea is as simple and concrete as it is effective. What if we went back to basics to gain a better grasp of the concept's power?
Sluse's algorithm: a forerunner of differential calculus | Tangente
Sluse, a 17th-century Belgian mathematician, devised an algorithm for systematically calculating the slope of the tangent to an implicitly defined algebraic curve. His work was a crucial step towards the discoveries of Newton and Leibniz.
The derivative in four stages: Fermat to Cauchy | Tangente
Mathematical concepts emerge gradually: they are often developed in several stages. The derivative is no exception. American historian Judith Victor Grabiner identifies four stages in its development.
Differentiation in nonstandard analysis
In nonstandard analysis, the derivative is introduced without invoking the concept of a limit. The price is that we must use infinitesimals: nonzero numbers whose absolute value is less than every strictly positive real number!
Local to global: the carefree theorem | Tangente
When faced with an optimization problem, we seek an extremum of a function.
A deep connection with determinism
Many physical and economic phenomena are described by differential equations. This implies, in particular, that the functions we seek are differentiable, but also that the phenomena concerned are locally deterministic.
Income tax and convex functions | Tangente
Tax due is a function of taxable income. What are its theoretical properties? Do the conclusions hold for the new property wealth tax? As it turns out, tax and derivatives make a fine pair!