These approximations can be generalized by using the derivative \( f \) to arbitrary order so that we have $$ \begin{equation} y_{i+1}=y(t=t_i+h)=y(t_i) + h(f(t_i,y_i)+\dots f^{(p-1)}(t_i,y_i) \frac{h^{p-1}}{p!}) + O(h^{p+1}). \tag{25} \end{equation} $$ These methods, based on higher-order derivatives, are in general not used in numerical computation, since they rely on evaluating derivatives several times. Unless one has analytical expressions for these, the risk of roundoff errors is large.