If you have a linear force-extension graph for say a spring then the spring constant is simply the gradient of the graph. However, how would you calculate the spring constant at a particular point on a non-linear (curved) graph for say an elastic band? Let's imagine you wanted the spring constant at 4cm of extension which corresponds to a load force of 4N. Would you simply do the ratio of the force and extension (i.e. one divided by the other) so 1N/cm in this case or would you find the gradient at that point (i.e. gradient of the tangent at that point)?
I would take data for force vs. extension for several data points (e.g., 5 or 6 data points), then plot them up. When using a package such as Excel, you can often find the best equation through the data points.
Assuming that the standard curve fits are not satisfactory, it is possible to surmise the form of the equation, then use the Solver add-in to tell you the constants involved. For example, if the form of the equation is $F=kx^n$, where "k" is the spring constant, "x" is the spring stretch, and "n" is not equal to 1, it is possible to set this equation up such that it looks like a least-squares functional form, which Solver is very good at solving. Simply rearrange this equation such that it becomes $(F-kx^n)^2=$ a residual, and calculate this residual for each data point, based on assumed values of "k" and "n". Then sum all of these residuals up, and tell Solver to manipulate "k" and "n" in order to minimize the sum of residuals. It will do so, and the resulting values of "k" and "n" will be the values that minimize the sum of the residual terms, which is actually minimizing the sum of the squared error terms. This answer will be the best equation through your data.