## Second Order Partial Derivatives

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# Second Order Partial Derivatives

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 ∂

(ii) ∂/∂y (∂z/∂x), denoted by ∂

(iii) ∂/∂x (∂z/∂y), denoted by ∂

(iv) ∂/∂y (∂z/∂y), denoted by ∂

Similarly, to find ∂

Usually, but not always a

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(i) ∂/∂x (∂z/∂x), denoted by ∂

^{2}z/∂x^{2}or f_{xx}.(ii) ∂/∂y (∂z/∂x), denoted by ∂

^{3}z/∂ydx or f_{xy}.(iii) ∂/∂x (∂z/∂y), denoted by ∂

^{2}z/∂x∂y or f_{xz}.(iv) ∂/∂y (∂z/∂y), denoted by ∂

^{2}z/∂y^{2}or f_{yy}.**Note**. To find ∂^{2}z/∂x^{2}(respectively ∂^{2}z/∂y^{2}), we take two successive derivatives, each time treating y (respectively x) as a constant.Similarly, to find ∂

^{2}z/∂y ax (respectively ∂^{2}z/∂y^{2}), 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 ∂^{2}z/ax ∂y and ∂^{2}z/∂y ax are called mixed or cross partial derivatives.Usually, but not always a

^{2}z/∂x∂y = ∂^{2}x/∂y ax. That is, the order of partial differentiation is immaterial. We assume that this is the case for all functions that we consider.For more help in

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