Partial derivative online calculator

Partial derivative concept is only valid to the multivariable functions. Examine two variable function z=f( x,y). Partial derivative by variables x and y are denoted as and correspondingly. The partial derivatives and by themselfs are also the two variable functions: and , so their partial derivatives can also be found:

Derivatives and are the second order partial derivatives of the function z by the variables x and y correspondingly. Derivatives and are called mixed derivatives of the function z by the variables x, y and y, x correspondingly. If the function z and their mixed derivatives and are defined at some neighborhood of a point M(x₀,y₀) and continuous at that point, then the following equality is valid:

Similary, one can introduce the higher order derivatives, for instance means that we should differentiate the function z two times by the variable x and three times by the variable y, so:

Sometimes, in order to denote partial derivatives of some function z=f(x,y) notations f_x'(x,y) and f_y'(x,y) are used. Subscript index is used to indicate the differentiation variable. Using this approach one can denote mixed derivatives: f_xy''(x,y) and f_yx''(x,y) and also the second and higher order derivatives: f_xx''(x,y) and f_xxy'''(x,y) accordingly. Following notations are equivalent:

, however, usually digit 1 is absent. To denote partial derivatives in our online calculator, we use symbols: . Example of step by step solution can be found here.