The metatheorem of calculus is that almost any function looks like a straight line when we zoom in far enough.
When we have zoomed in far enough we cannot tell the difference between the graph and the tangent line.
If we have zoomed in that far we can approximate the tangent line with a secant line.
The slope of that line is the derivative at that point.

This applet zooms in on y=f(x) around the point A.
A is a draggable point.
The box extends out from A by a distance of zoom up and down, right and left.
In the zoom window you can also see the secant line using A and a point zoom/2 to the right of A.