Partial derivatives

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, fx(x,y), etc.

∂z/∂x = lim [f(x + δx,y) - f(x,y)]/δx
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, fy(x,y), etc.

∂z/∂y = lim [f(x,y + δy) - f(x,y)]/δy
Notation; ∂z/∂x = p, ∂z/∂y = q, ∂2z/∂x2 = r, ∂2z/∂x∂y = s, ∂2z/∂y2 = t

Partial Derivatives of higher order

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.

∂(∂z/∂x)/∂x = ∂2z/∂x2 or ∂2f/∂x2 or fxx
∂(∂z/∂x)/∂y = ∂2z/∂y∂x or ∂2f/∂y∂x or fyx
∂(∂z/∂y)/∂x = ∂2z/∂x∂y or ∂2f/∂x∂y or fxy
∂(∂z/∂y)/∂y = ∂2z/∂y2 or ∂2f/∂y2 or fyy
2z/∂y∂x = ∂2z/∂x∂y