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I am looking for a simple direct proof of the fact that $$ \frac{\frac{f(x + \Delta x) - f(x)}{\Delta x} -f'(x)}{\Delta x} \stackrel{\Delta x \to 0}{\to} \frac{1}{2}f''(x), $$ or, equivalently, $$ f(x+\Delta x) = f(x) + f'(x)\Delta x + \frac{1}{2}f''(x)\Delta x^2 + o(\Delta x^2). $$

holds for a twice-differentiable $f(x)$.

I remember there were times when I could derive this directly from the following definition of a derivative: $$ f(x+\Delta x) = f(x) + f'(x)\Delta x + o(\Delta x) $$ in a couple of simple lines.

A long time has passed since then and now I need to either recollect this magical "obvious" proof of mine or find out I was wrong then and the actual proof is more involved.

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