If z = f (x,y), then not only is z a function of x and y, but also ∂z/∂x and ∂z/∂y are each functions of x and y. Thus we can form partial derivatives of ∂z/∂x and ∂z/∂y as we formed those of  z to obtain the following second-order partial derivatives of z:

(i)    ∂/∂x (∂z/∂x), denoted by ∂²z/∂x² or f_xx.

(ii)    ∂/∂y (∂z/∂x), denoted by ∂³z/∂ydx or f_xy.

(iii)   ∂/∂x (∂z/∂y), denoted by ∂²z/∂x∂y or f_xz.

(iv)   ∂/∂y (∂z/∂y), denoted by ∂²z/∂y² or f_yy.

Note . To find ∂²z/∂x² (respectively ∂²z/∂y²), we take two successive derivatives, each time treating y (respectively x) as a constant.

Similarly, to find ∂²z/∂y ax (respectively ∂²z/∂y²), we take two successive derivatives, whereas, in the first differentiation y (respectively x) is treated as a constant and in the second differentiation x (respectively y) is treated as a constant.

Note. The derivatives ∂²z/ax ∂y and ∂²z/∂y ax are called mixed or cross partial derivatives.

Usually, but not always a²z/∂x∂y = ∂²x/∂y ax. That is, the order of partial differentiation is immaterial. We assume that this is the case for all functions that we consider.

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