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 $O(h^{2})$.