\partial_{e_2}f(T_0)=\lim_{t \rightarrow 0} \frac{ f(T_0+te_2)-f(T_0)}{t}=\lim_{t \rightarrow 0} \frac{ f(0,t+y_0,0)}{t}=\lim_{t \rightarrow 0} \frac{0}{t}=0.