proof of finding derivative

suppose we have a function $f(x)$

we need a mathematical way to find a function that gives us the instantaneous rate of change for every value of the function, which is the derivative of the function. how can we find this?

we need to break this down. firstly, we know how to find the average rate of change over a domain interval.

for example, if wanted to know the average rate of change in the interval $x\in [2,7]$, we just use $\frac{f(7)-f(2)}{7-2}$

to put this into general form,