*Partial derivative* concept is only valid for multivariable functions. Examine two variable function
z=f(x,y).
Partial derivative by variables
and
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
by the variables
and
correspondingly. Derivatives
and
are called mixed derivatives of the function
by the variables
,
and
,
correspondingly. If the function
and their mixed derivatives
and
are defined at some neighborhood of a point
M(x_{0},y_{0})
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 two times by the variable and three times by the variable 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:

To denote partial derivatives in our online calculator, we use symbols:
;
;
.
Sample of step by step solution can be found
here.

Partial derivative calculator

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