\partial_{e_3}f(T_0)=\lim_{t \rightarrow 0} \frac{ f(T_0+te_3)-f(T_0)}{t}=\lim_{t \rightarrow 0} \frac{ f(0,y_0,t)}{t}=\lim_{t \rightarrow 0} \frac{0}{t}=0.