Changes

==Definition==

==Definition==

The derivative of the function <tt>f(x)</tt>, <tt>f'(x)</tt> or <math>\frac{df}{dx}</math>

The derivative of the function <tt>f(x)</tt>, denoted <tt>f'(x)</tt> or <math>\frac{df}{dx}</math>, is defined as:

 

is defined as:

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

and so forth.

and so forth.

A common alternative notation is <math>f''(x)</math>, <math>f'''(x)</math>, and <math>f^{(n)}(x)</math>.

A common alternative notation is <math>f''(x)</math>, <math>f'''(x)</math>, and <math>f^{(n)}(x)</math> for the second, third or n th derivative.

===Partial derivatives===

===Partial derivatives===

*<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>

*<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, specifically continuous 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, specifically continuous functions.

==Uses==

==Uses==