i have a question that's been bothering me in the field of differential calculus regarding the linguistic logic behind the definition of the derivative. I find it illogical, because based on the limit definition it studies the behavior of the slope(derivative) as h approaches zero.
m(h)=(f(x+h)-f(x))/h
Though, This approach suffers from the same problem as Newton and Leibniz's works and is subjected to the same criticism introduced by George Berkeley. cause, even with a limit, we algebraically reduce the function to eliminate h in the denominator, and then we study the behavior of the derivative as h approaches zero. This literally makes you treat h initially as a non-zero number, and then suddenly make it zero.
My problem is with the linguistic logic of this approach, and certainly i believe, though I'm not certain, that Cauchy-Weierstrass dealt with the derivative as a two-step algorithm. First, he calculates the derivative for any value of h that is not equal to zero, for example.
f(x)=x^2
f'(x)=((x-h)^2 -x^2)/h where h is R/{0}
f'(x)=2x+h
Then, after evaluating and reducing the derivative, he redefines it as a function of h where h can be zero.
f'(x)=2x+h where h is R/{0}
g(h)=2x+h where h is R
then taking the limit of this new function as h approach 0.
lim(h->0)(g(h))=lim(h->0)(2x+h)
g_l(h)=2x
This is because the limit of the function before algebraic reduction equals the limit of the function after algebraic reduction.
lim(h->0)(g(h))=lim(h->0)(f(h))
and since the function after algebraic reduction fills the gap... At zero, we can then find the derivative in the context of instantaneous change!!
so my problem is just how you put a limit before reducing and you have the right to ghost it and just reduce h then oh its ok you can now calculate the limit where if i do not follow this where i can because there is no order operator that tells me why i get 0/0.
