I had a thought and I am not sure how to answer it. Let's say that I have some data points (x1,y1), (x2,y2), (x3,y3) ... and I want to estimate the slope at x2. Would it be better to estimate it using (y2 - y1)/(x2 - x1) or (y3 - y1)/(x3 - x1) ? That is, should the secant line that I draw have y(x2) as an endpoint or contain y(x2)? I feel like the latter would better describe the behavior since it is possible that drastic things could be happening AT y(x2) and so by drawing the secant through it, we get a better picture of the overall behavior around y(x2). What do you think?