03/02/2019, 08:13 PM
Dear Jay Fox,
Thankyou so much for your replies and interest! I also need to spend some time working with the pathway that you provided to see if I can recreate your derivation, it is probably correct, I would just like to also be sure I can follow your derivation. Since I originally posted this, it occurred to me that the \(\partial f\) defined in the geometric derivative and the further derivatives is definitely not isomorphic to the original differential \(\partial f\). Consider the following.
Then we must draw a direct relationship between the two \(\partial f\)'s in order to re-specify the original postings formula properly. We can see that clearly,
Further note that because the definition of the real logarithm function is not continuous at 0 (it is asymptotic), the limit and logarithm are not interchangeable, hence to generalize the expression of \(\partial f^{\prime}\), however, the argument of the log in this case is neither zero, nor a function of zero (unless at f(x) = 0 where everything breaks down) hence we must be able to specify an invertable functional form g(.) such that:
of equivalently,
Thankyou so much for your replies and interest! I also need to spend some time working with the pathway that you provided to see if I can recreate your derivation, it is probably correct, I would just like to also be sure I can follow your derivation. Since I originally posted this, it occurred to me that the \(\partial f\) defined in the geometric derivative and the further derivatives is definitely not isomorphic to the original differential \(\partial f\). Consider the following.
\( \lim\limits_{\Delta x \to 0} ( f(x + \Delta x) - f(x) )=\partial f\qquad\qquad (1) \)
\( \lim\limits_{\Delta x \to 0} \left(\frac{f(x+\Delta x)}{f(x)}\right) =\partial f^{\prime}\qquad\qquad (2) \)
Then we must draw a direct relationship between the two \(\partial f\)'s in order to re-specify the original postings formula properly. We can see that clearly,
\( \lim\limits_{\Delta x \to 0}\ln\left(\frac{f(x+\Delta x)}{f(x)}\right)=\lim\limits_{\Delta x \to 0}(\ln{f(x + \Delta x)} - \ln{f(x)}) \ne \lim\limits_{\Delta x \to 0}\ln{f(x+\Delta x) - f(x)}\qquad\qquad (3) \)
Further note that because the definition of the real logarithm function is not continuous at 0 (it is asymptotic), the limit and logarithm are not interchangeable, hence to generalize the expression of \(\partial f^{\prime}\), however, the argument of the log in this case is neither zero, nor a function of zero (unless at f(x) = 0 where everything breaks down) hence we must be able to specify an invertable functional form g(.) such that:
\( g(\ln f(x + \Delta x) - \ln f(x)) = \ln(f(x + \Delta x) - f(x))\qquad\qquad (4) \)
of equivalently,
\( \ln f(x + \Delta x) - \ln f(x) = g^{-1}(\ln(f(x + \Delta x) - f(x)))\qquad\qquad (5) \)