On Mon, Dec 02, 2002 at 08:46:32PM +0000, Earle Martin wrote: > and in general: > > x + x + ..... + x = x^2 > \___ x times ___/ > > Derive in place: > > 1 + 1 + ..... + 1 = 2 x > \___ x times ___/ > > 1 * x = 2 x
If you're going to do differentiate a function of a function in x you will another stage. The "x times" is obviously a function of x. Let f = u(v(x)). u(v) = v^2; v(x) = x Using the chain rule, du = du . dv = 2v . 1 = 2x dx dv dx > Note the formula is valid for pure fractionals too, for example: > > 0.05 + 0.05 + 0.05 + 0.05 + 0.05 = > 0.01 * (5 + 5 + 5 + 5 + 5) = 0.01 * 5^2 = 0.05 ^2 = 0.25 > > and thus it is valid for any real number x because we can always write x as > a sum of a pure integer u and a pure fractional v: x = u+v. For example > x=3.8 => u=3 and v=0.8. So we are not deriving in a discrete space, think > somewhere else. This is totally spurious. Fear the rigor of proofs that say "derive" instead of "differentiate" and "not in a discrete space, somewhere else" :-) Paul -- Paul Makepeace ....................................... http://paulm.com/ "If I could have what I want, then feeble screebles." -- http://paulm.com/toys/surrealism/
