Derivatives

Reconstruct our notion of derivative given these defnitions and results and a new understanding of the limit notion.

.    If F converges then it converges uniquely

     Suppose it converges to b, then if i Î I \ {0}, b = st(F*(a + i))

     We call this the lim_x®a F(x)

.    Now, for F: R ® R, a Î R, the conventional definition goes:

            F'(a) = lim_h®0 [F(a + h) - F(a)] / h

     Using the new definition of a limit

            F'(a) = b iff "dx Î I \ {0} we have

            dF/dx » b (for dF = F*(a + dx) - F(a))

     If b exists - if F'(a) exists - then

            F'(a) = st(dF/dx)             

Example    F(x) = x²

                 dF/dx = [(a + dx)² - a²] / dx

                          = [2a(dx) + (dx)²] / dx 

                          = 2a + dx

                          = 2a