Let z = f(x,y) be function of two independent variables x and y. If we keep y constant and x varies then z becomes a function of x only. The derivative of x with respect to x, keeping, y as constant is called partial derivative of "z", w.r.t. "x" and is denoted by symbols

∂z/∂x, ∂f/∂x, f

_{x}(x,y), etc.

Then,

*∂z/∂x = lim [f(x + δx,y) - f(x,y)]/δx
δx→0*

The process of finding the partial differential coefficient of z w.r.t. 'x' is that of ordinary differentiation, but the only difference that we treat y as constant.

Similarly, the partial derivative of 'z' w.r.t. 'y' keeping x as constant is denoted by

∂z/∂y, ∂f/∂y, f

_{y}(x,y), etc.

Then,

*∂z/∂y = lim [f(x,y + δy) - f(x,y)]/δy
δy→0*

Notation; ∂z/∂x = p, ∂z/∂y = q, ∂

^{2}z/∂x

^{2} = r, ∂

^{2}z/∂x∂y = s, ∂

^{2}z/∂y

^{2} = t

Let z = f(x,y), then ∂z/∂x and ∂z/∂y being the functions of x and y can further be differentiated partially with respect to x and y.

Symbolically

∂(∂z/∂x)/∂x = ∂^{2}z/∂x^{2} or ∂^{2}f/∂x^{2} or f_{xx}

∂(∂z/∂x)/∂y = ∂^{2}z/∂y∂x or ∂^{2}f/∂y∂x or f_{yx}

∂(∂z/∂y)/∂x = ∂^{2}z/∂x∂y or ∂^{2}f/∂x∂y or f_{xy}

∂(∂z/∂y)/∂y = ∂^{2}z/∂y^{2} or ∂^{2}f/∂y^{2} or f_{yy}

∂^{2}z/∂y∂x = ∂^{2}z/∂x∂y