\partial_{e_1}f(T_0)=\lim_{t \rightarrow 0} \frac{ f(T_0+te_1)-f(T_0)}{t}=\lim_{t \rightarrow 0} \frac{ f(t,y_0,0)}{t}=\lim_{t \rightarrow 0} \frac{-t^3}{t^3}=-1.