Instantaneous rate of change?!

I have a table of values of X and Y, so how do I find the instantaneous rate of change at a specific X value without graphing?

Technically, you cannot. You do not have sufficient data. Because you have a set of discrete points, you can know the average rate of change between any two "x" points. The instantaneous rate of change at any particular "x" may be higher or lower than the average.

Now if you choose to fit a particular curve (function) to the data you have, then you can calculate the instantaneous rate of change of that curve (usually by differentiating).

If you want to find the average rate of change between any two "x" in your table, then you just take the change in y over the change in x.

(y1 - y0) / (x1 - x0)