The slope taken is linearized version of equations presented in the model. Linearization usually is implemented by removing non-linear parts: instead of those their differential expressions were to be used $f(x) \approx f(a) + f'(a)(x-a)$. That is shown on the picture. That differential $f'(a)$ in question could be computed by interpolation techniques, which is separate (complex) task. Then you solve the equation, with usual methods (linear algebra approach). Then after solution is found, again you recompute all differentials. And so on.
But they are also use the confusing words. Namely "forces". Forces are actually used, but in numerical methods it is special "forces", in optimization algorithms, where no actual forces are present. And existing "exact" system is substituted by differential system with moving "points", for example so-called "model of heavy ball".
If they are solving optimisation problem, for example find the point of balance in system, then do not misinterpret these "forces" with actual mechanical forces.
Also note, that in special applications there is no much need in knowing the stability region of the system, but to understand the borders of this region. So optimization/nonlinear solving algorithms could introduce "energy pumping" into the system, to find out the points of maximum fluctuations. Of cause it may be chosen to (or may not) be analogous to actual Newton laws governing the mechanical equations. That "pumping" also does not have anything to do with actual physics, but is a set of methods having its roots in theory of dynamical systems.