Exercise 29

Suppose that a function $f = f(x,y)$ is differentiable at a point $(x_0,y_0)$. Let $L = L(x,y) = f(x_0,y_0) + m(x-x_0) + n(y-y_0)$ as in the conditions of Definition. Show that $m = f_x(x_0,y_0)$ and $n = f_y(x_0,y_0)$. (Hint: Calculate the limits of the relative errors when $h = 0$ and $k = 0$.)