I love calculus.
If we were to measure the derivative of which is the slope at certain point where tangent line touch the graph, we have to draw another line that seems to touch the point very close to it.
Lets pay our attention to the graph above. We see that tangent line (the red one) touch the graph and secant line (the dashed blue one) is adjusted, so we can approximately measure the derivative of .
If (the gap tends to 0), both lines will be merged as if they are nothing but one single straight line. This is the main idea of how we measure derivative using geometrical approach. Now, we can only see the gap between tangent and secant line using highly modern microscope.
In high school, we’ve encountered the formula to measure the slope . means slope, means the change in axes, means the change in axes.
Take that is the red line and is the dashed blue line, we will have 2 point and .
Because derivative of is a mere slope, we can get this formula right off the bat: