how to find slope of discrete point?

Question

I am wondering if it is possible to find the slope at each point in the following dataset,

% X Y %=================== 0.7761 0.5715 0.794 0.5729 0.8117 0.5744 0.8292 0.5762 0.8465 0.5782 0.8637 0.5804 0.8807 0.5828 0.8977 0.5853 0.9144 0.5879 0.9311 0.5907 0.9477 0.5937 0.9641 0.5968 0.9805 0.6 0.9967 0.6033 1.0129 0.6067

I understand that the slope can be obtained using the difference of the two neighboring points by

$$m = \frac{y_2-y_1}{x_2-x_1}$$

and, the angle that each point made with the $x$-axis is essentially the $atan$ of $m$

$$\theta = \tan^{-1}(m) $$

But, is it possible to calculate the slope without using the above formula? without trying to curve-fit the points.

Answer 1

In my experience, when trying to estimate the slope at a point, it is better to use the slope of the line between the preceding and following point.

This is analogous to the fact that $f'(x)$ is more accurately estimated by $(f(x+h)-f(x-h))/(2h)$ (error of order $h^2$) than by $(f(x+h)-f(x))/(h)$ (error of order $h$).