The partial derivative of a multivariable function is simply its derivative with respect to only one variable, keeping all other variables constant (which are not functions of the variable in question). The formal definition is
where is the standard basis vector of the th variable. Since this only affects the th variable, one can derive the function using common rules and tables, treating all other variables (which are not functions of ) as constants. For example, if , then
Note that in equation , we treated as a constant, since we were differentiating with respect to . The partial derivative of a vector-valued function with respect to variable is a vector .
Multiple partial derivatives can be treated just like multiple derivatives. There is an additional degree of freedom though, as you can compound derivatives with respect to different variables. For example, using the above function,
is another way of writing . If is continuous in the neighborhood of ,
and and are continuous in an open set , it can be shown
(see Clairaut’s theorem (http://planetmath.org/ClairautsTheorem)) that in , where are the ith and jth variables. In fact, as long as an equal number of partials are taken with respect to each variable, changing the order of differentiation will produce the same results in the above condition.
Another form of notation is where is the partial derivative with respect to the first variable times, is the partial with respect to the second variable times, etc.
|Date of creation||2013-03-22 11:58:30|
|Last modified on||2013-03-22 11:58:30|
|Last modified by||Mathprof (13753)|