Find the Taylor series of a function of two variables, check my answer.

Question

I want to find Taylor expantion of the function with two variables $f(x_i+p_1h,y_i+q_1k_1h)$ .

If $y'_i=f(x_i,y_i)=f_i$, \begin{eqnarray} f(x_i+p_1h,y_i+q_1k_1h)&=&f(x_i,y_i)+p_1hf_x+q_1k_1hf_y\nonumber\\ &=&f_i+p_1hf_x+q_1k_1hf_y \end{eqnarray}

Please check my answer. Is it correct or incorrect?