Okay, I was wondering if there was a way to take a sortcut to the second derivitive of a function? What I'm talking about here directly involves the differnce quotient. I know all the little rules of differentiation that are derived from the difference quotient, and I know how to use them to find the second derivitve pretty easily. But I'm talking about something more "theoretical", and less "applicable". Let me start by using this definition:

$\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Now, lets say we are not going to use any of the fancy simple shortcuts that have been proved about diferentiation; lets say that we are required to use the difference quotient, can we find a short cut some how? Heres what I mean:

We start with the definition above, from that we can get:

$\displaystyle f''(x) = \lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h}$

Now, expanding things gives:

$\displaystyle f''(x) = \lim_{h \to 0} \frac{\lim_{h \to 0} \frac{f((x+h)+h) - f(x+h)}{h} - \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}}{h}$

My question is, can we algebraically manipulate the expression above to get a different looking difference quotient? Or some other sort of limit expression that would be a 'short-cut' to the second derivitive? Some function involving $\displaystyle f(x)$ that we take the limit of as $\displaystyle h \to 0$? (I'm gona call this function $\displaystyle g(x)$). So that we can end up with a simpler form of the limit above that takes us striaght to the second derivitive? Something of the form:

$\displaystyle f''(x) = \lim_{h \to 0} g(x)$

?

Thanks in advance