Answer
See explanation
Work Step by Step
We know that the derivative of a function at a point is the slope of the tangent line to the function at that point.
Let the point under consideration be $A(x, f(x))$.
Now, in order to find the slope, we will consider two other points $h$ distance away from $A$, where $h$ is an infinitesimally small number. (taking $\lim\limits_{h \to 0}$)
We also know the formula for the slope $m$ of a graph:
$m = \frac{y_2-y_1}{x_2-x_1}$
Using the two points on either side of the point $A$, $(x-h, f(x-h))$ and $(x+h,f(x+h))$, we have:
$\implies m = \frac{f(x+h)-f(x-h)}{(x+h)-(x-h)} = \frac{f(x+h)-f(x-h)}{2h}$
Adding limit so that the infinitesimal distance is accounted for:
$\therefore m = \lim\limits_{h \to 0}\frac{f(x+h)-f(x-h)}{(x+h)-(x-h)}$
This gives an accurate approximation of the value of the derivative at point $A$.
