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From his article: You can calculate the derivative, or rate of change, of a function f by doing (f(x+ε) - f(x)) / ε and then at the end throwing out terms in
by doctorpangloss 12y ago
From his article:
You can calculate the derivative, or rate of change, of a function f by doing
(f(x+ε) - f(x)) / ε
and then at the end throwing out terms involving ε
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To be clear, this formula is typically introduced as the finite difference formula in calculus instruction.
But differentiation of polynomials is straightforward, so don't be too impressed that a simple formula finds the derivative of x-squared in his example. A bunch of simple procedures find the derivatives of polynomials.
For more complicated algebraic functions, like rational functions, nearly every calculus student is taught a collection of shortcuts that are fundamentally taking the limit of the finite difference formula as epsilon (or "h" commonly) approaches zero.
- noelwelsh 12y agoI think you've missed the point. You're correct that one can do basic calculus using a bunch of ad-hoc shortcuts, and it is commonly taught this way. Also, as you probably know, calculus was first formalised using limits not infinitesimals. However this theory of infinitesimals is not ad-hoc. It is a complete and rigorous alternate formulation of calculus in terms of an extension of the reals, known as the hyperreals, that includes infinitesimals. This allows computation in a principled way that matches the intuitions on which beginning calculus is often taught. It's pretty cool stuff, and worth reading about even if you never use it in practice.