The partial derivative only acts on one the variables on the equations and treats the others as constant.

The spacial derivative is the measure of a quantity as and how it is being changed in space. This is different from a temporal derivative and partial derivative.

A partial derivative is the derivative of a function of more than one variable with respect to only one variable. When taking a partial derivative, the other variables are treated as constants. For example, the partial derivative of the function f(x,y)=2x2 + 3xy + y2 with respect to x is:
?f/?x = 4x + 3y
here we can see that y terms have been treated as constants when differentiating.

The partial derivative of f(x,y) with respect to y is:
?f/?y = 3x + 2y
and here, x terms have been treated as constants.

what are the applications of partial derivative in real analysis.

Suppose, Z is a function of X and Y. In case of Partial Differentiation of Z with respect to X, all other variables, except X are treated as constants. But, total derivative pf z is given by,

dz=(partial derivative of z w.r.t x)dx + (partial derivative of z w.r.t y)dy