4. Higher Order Parital Derivatives

Second Order Partial Derivatives

Notation:

$(f_x{)}_x=f_{xx}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial x}\right)=\frac{{\partial }^{2}f}{\partial x^2}=\frac{{\partial }^{2}z}{\partial x^2}$

$(f_x{)}_y=f_{xy}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{{\partial }^{2}z}{\partial y\partial x}$

$(f_y{)}_x=f_{yx}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)=\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}z}{\partial x\partial y}$

$(f_y{)}_y=f_{yy}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial y}\right)=\frac{{\partial }^{2}f}{\partial y\partial y}=\frac{{\partial }^{2}z}{\partial y\partial y}$

Higher-Order Derivatives

Notation

Consider the function $f(x,y,z)$:

$f_{zyxy}=(((f_z{)}_y{)}_x{)}_y=\frac{\partial }{\partial y}\left(\frac{\partial }{\partial x}\left(\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial z}\right)\right)\right)=\frac{{\partial }^{4}f}{\partial y\partial x\partial y\partial z}$

Clairaut’s Theorem

Theorem

If the function $f$ has continuous second-order partial derivatives, then:

$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$

Remark: The theorem can be generalized to higher-order derivatives, and to functions of more than two variables.