Overview
{% f(x_0 + \Delta x) - f(x_0) = f'(x_0)\Delta x + \frac{1}{2} f''(x_0) \Delta x ^2 + Remainder %}
Then, if we take multiple such steps, such that {% x-x_0 = \sum \Delta x %}
{% f(x) - f(x_0) = \sum [f'(x_0)\Delta x + \frac{1}{2} f''(x_0) \Delta x ^2 + Remainder] %}
Then as {% \Delta x \rightarrow 0 %}, this becomes
{% f(x) - f(x_0) = \int f'(x_0)\Delta x %}
This is true because both {% \sum \Delta x^2 + Remainder \rightarrow 0 %}
However, in the case that {% x %} is an Borwnian Motion {% \sum \Delta x^2 \rightarrow t %}, the quadratic variation.