Partial Derivatives of Higher Order

If z= f(X,y), then partial derivatives of z with respect to x and y, will be ∂z/∂x and ∂z/∂y respectively, and again, these are, in general, the functions of x and y. Thus, we can process partial derivatives of ∂z/∂x and  ∂z/∂y just as we formed those of z to obtain the following second-order partial derivatives of z.

(i)  = ∂/∂x (∂z/∂x) =∂²z /∂x² = ƒ××      (ii)   = ∂/∂y(∂z/∂x) =∂²z /∂y∂x = ƒy×

(iii)   = ∂/∂x(∂z/∂x) =∂²z /∂y∂x = ƒxy   (iv)   = ∂/∂x(∂z/∂y) =∂²z /∂y²z= ƒxy

In order to find out ∂²z /∂y² (and ∂²z /∂y² ), we have to take two successive derivatives, each time treating y and x as a constant respectively. Similarly, to find ∂²z /∂y∂x (and ∂²z /∂y∂x), we have to take two successive derivatives, whereas in the first differentiation y (respectively) x us treated as a constant and in the second differentiation of x (respectively) y is treated as a constant. The derivatives ∂²z /∂x∂y and ∂²z /∂y∂x are called mixed or cross partial derivatives.