With all this talk of approximating derivatives, we take it for granted that derivatives work.  Somewhat surprisingly perhaps, this hasn’t always been the case.  Derivatives present some odd philosophical issues.  Newton’s original method of differential calculus required dividing by a number that’s zero in all but name; no doubt many of us remember how well that worked out in high school algebra.  From the discoveries of Newton and Leibniz until Karl Weierstrass formalized the limit in the 19th century, the difference equation definition of the derivative required use of these pesky infinitesimals.Infinitesimals bothered quite a few philosophers and mathematicians (engineers, of course, were no doubt apathetic).   One of the philosophical fads at the time, Empiricism, was expounding the use of evidence in constructing models of the world; basically, that all human knowledge should be derived from observations of the world.  To the Empiricists (John Locke, David Hume, and George Berkeley to name a few), therefore, the idea of using a number so small it cannot be seen posed some interesting issues– not helped by the derivative’s success at modeling the universe.  Some clergy members even considered calculus a bit of a cult for its belief in senseless (an decidedly non-Aristotelean) math.The limit is nice in that it allows the definition of the derivative to be written in terms of ordinary numbers that keep getting smaller.  Mathematicians have largely removed infinitesimals from the rigorous  definitions of integrals and derivatives, although engineers and physicists still use them on occasion. There’s a decent treatment of this in Zero: The Biography of  Dangerous Idea by  Charles Seife http://en.wikipedia.org/wiki/Weierstrasshttp://en.wikipedia.org/wiki/Empiricismhttp://www.friesian.com/calculus.htm