# 14.7 | Second-Order Partial Derivatives

If $$f_{xy}$$ and $$f_{yx}$$ are continuous at $$(a,b)$$, an interior point of their domain, then
$$f_{xy}(a,b) = f_{yx}(a,b).$$
The Taylor polynomial of degree 1 approximating $$f(x,y)$$ for $$(x,y)$$ near $$(a,b)$$ is given by
$$L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b).$$
The Taylor polynomial of degree 2 approximating $$f(x,y)$$ for $$(x,y)$$ near $$(a,b)$$, provided that $$f$$ has continuous second-order partial derivatives, is given by
$$L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)+ \frac{f_{xx}}{2}(a,b)(x-a)^2 + f_{xy}(a,b)(x-a)(y-b) + \frac{f_{yy}}{2}(a,b)(y-b)^2.$$