Changes

Note that the two partial derivatives <math>f_1(x_1,x_2)</math> and <math>f_2(x_1,x_2)</math> in this  example are again differentiable functions of <math>x_1</math> and <math>x_2</math>, so higher derivatives can be calculated:

Note that the two partial derivatives <math>f_1(x_1,x_2)</math> and <math>f_2(x_1,x_2)</math> in this  example are again differentiable functions of <math>x_1</math> and <math>x_2</math>, so higher derivatives can be calculated:

*<math>f_{11}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_1}\;</math><math> = \frac{6 x_1}{1+x_2^2}</math>

*<math>f_{11}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_1} = \frac{6 x_1}{1+x_2^2}</math>

*<math>f_{12}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_2}\;</math><math> = \frac{- 6 x_1^2 x_2}{(1+x_2^2)^2}</math>

*<math>f_{12}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_2} = \frac{- 6 x_1^2 x_2}{(1+x_2^2)^2}</math>

*<math>f_{21}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_1}\;</math><math> = \frac{- 6 x_1^2 x_2}{(1+x_2^2)^2}</math>

*<math>f_{21}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_1} = \frac{- 6 x_1^2 x_2}{(1+x_2^2)^2}</math>

*<math>f_{22}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_2}\;</math><math> = \frac{8 x_1^3 x_2^2 -2 x_1^3 -2 x_1 x_2^2}{(1+x_2^2)^3}</math>

*<math>f_{22}(x_1,x_2) = \frac{\partial^2 f(x_1,x_2)}{\partial x_2 \partial x_2} = \frac{8 x_1^3 x_2^2 -2 x_1^3 -2 x_1 x_2^2}{(1+x_2^2)^3}</math>

Note that <math>f_{12}(x_1,x_2)</math> equals <math>f_{21}(x_1,x_2)</math>, so that the order of taking the derivative doesn't matter. Though this doesn't hold generally, it's true for a great class of important functions.

Note that <math>f_{12}(x_1,x_2)</math> equals <math>f_{21}(x_1,x_2)</math>, so that the order of taking the derivative doesn't matter. Though this doesn't hold generally, it's true for a great class of important functions.