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
$$
L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)+ \frac{f_{xx}}{2}(a,b)(x-a)^2 + f_{xy}(a,b)(x-a)(y-b) + \frac{f_{yy}}{2}(a,b)(y-b)^2.
$$

E 14.7 Exercises