Let us then include the second derivative in our Taylor expansion. We have then $$ \begin{equation} \Delta(t_i,y_i(t_i))=f(t_i)+\frac{h}{2}\frac{df(t_i,y_i)}{dt}+O(h^3). \tag{22} \end{equation} $$ The second derivative can be rewritten as $$ \begin{equation} y''=f'=\frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial y}f \tag{23} \end{equation} $$ and we can rewrite Eq.\ (14) as $$ \begin{equation} y_{i+1}=y(t=t_i+h)=y(t_i) +hf(t_i)+ \frac{h^2}{2}\left(\frac{\partial f}{\partial t}+\frac{\partial f}{\partial y}f\right) + O(h^{3 }), \tag{24} \end{equation} $$ which has a local approximation error \( O(h^{3 }) \) and a global error \( O(h^{2}) \).