Am 03.03.2014 18:18, schrieb email@example.com: > On Mon, 03 Mar 2014 08:22:46 +0100, Roland Franzius > <firstname.lastname@example.org> wrote: > >> Am 02.03.2014 20:46, schrieb email@example.com: >>> >>> But we don't know the derivative! The question was how do you find the >>> derivative of sin(x) in the New Calculus. An answer that only works >>> once you know the derivative isn't very useful. >> >> It is the other way around. >> >> If one does not start with the inversion of the integral of dx/(1+x^2) >> or dx/sqrt(1-x^2) or with the differential equations f' =g , g'=-f for >> the trigonometric functions or with their taylor expansions, one has to >> deduce the derivatives of sin'(0) = 1 and cos'(0) = 0 from the geometric >> definitions. >> >> For the integrals of the inverse trigonometric functions and for the >> system f' = g , g'=-f these values evidently are boundary conditions. >> >> (sin(x+h)-sin(x)) = sin(x) (cos (h)-1)/h + cos(x) sin(h)/h >> >> Its trivial to show from the geometric definitions using some area >> majorants or half angle formulas that >> >> lim_h->0 (cos (h)-1)/h =0 >> and >> lim_h->0 sin(h)/h =1 >> >> In this way you can introduce exact division formulas for polynomials >> >> (f(sin,cos,tan)(x+h)-f(sin,cos,tan)(h)))/(sinh) >> >> using the trigonometric algebra. >> >> The same is true for log, ist inversion, the exponentials, general >> linearity and quotients and products and even the field of the complex >> elliptic functions as shown by Jacobi and Abel. >> >> There is a deeply hidden algebraic reason why calculus is called >> calculus and not limitus. The Cauchy-Weierstrass axiomatic foundation >> makes the calculus an exact logical body of consistency but seldom >> generates formulas. >> >> For their proofs most of the corpus of formulas of elementary functions >> and higher transcendental functions need only one or two essential limit >> theorems about absolutely, equicontinuous convergence for derivatives. > > Well of course. But I don't see how anything you say here has any > relevance to the thread. The question is how we determine the > derivative of sin(x) in the New Calculus - nothing above can > have any bearing on that, since what's above involves all > sorts of things that we are told do not exist in NC.
You are right, under these assumptions there do not exist nonrational functions. Consequently under these assumptions we have not the problem to derive sin but to define it.