One way is to create the function that could generate your points in the form of F(X)=some expression depending on X.
There are different ways to generate such functions, for example the least square methods (already built in Excel for example).
One you know the function, you could calculate the derivative at any point assuming that the function is continuous at that point.
Given the points (1,1) , (1,8) , (1, 27) you can see that F(X)=X*X*X and its derivative is 3*X*X so the derivative at x=1 is 3*1*1.
The problem here is that since there are many methods to obtain the original function, there will be a derivative for each generated function that may or may not all yield the same result.
There is another approach to calculate first derivative numerically without knowing the exact function - Please see: http://www.math.montana.edu/frankw/ccp/modeling/continuous/heatflow2/firstder.htm