Answer
$f_{xx}=k(k-1)x^{k-2}y^lz^{m}$
$f_{xy}=klx^{k-1}y^{l-1}z^{m}$
$f_{yx}=klx^{k-1}y^{l-1}z^{m}$
$f_{yz}=lmx^{k}y^{l-1}z^{m-1}$
$f_{xz}=kmx^{k-1}y^{l}z^{m-1}$
$f_{zx}=kmx^{k-1}y^{l}z^{m-1}$
$f_{yy}=l(l-1)x^ky^{l-2}z^{m}$
$f_{yz}=lmx^ky^{l-1}z^{m-1}$
$f_{zy}=lmx^ky^{l-1}z^{m-1}$
$f_{zz}=m(m-1)x^ky^{l}z^{m-2}$
Work Step by Step
Given: $f(x,y,z)=x^ky^lz^m$
Partial differentiate with respect to $x$ is $f_x=kx^{k-1}y^lz^m$ ... (1)
Partial differentiate with respect to $y$ is $f_y=lx^{k}y^{l-1}z^m$ ...(2)
Partial differentiate with respect to $z$ is $f_z=mx^{k}y^{l}z^{m-1}$ ...(3)
Partial differentiate equation (1) with respect to $x$ is
$f_{xx}=k(k-1)x^{k-2}y^lz^{m}$
Partial differentiate equation (1) with respect to $y$ is
$f_{xy}=klx^{k-1}y^{l-1}z^{m}$
Partial differentiate equation (2) with respect to $x$ is $f_{yx}=klx^{k-1}y^{l-1}z^{m}$
Partial differentiate equation (2) with respect to $z$ is $f_{yz}=lmx^{k}y^{l-1}z^{m-1}$
Partial differentiate equation (1) with respect to $z$ is $f_{xz}=kmx^{k-1}y^{l}z^{m-1}$
Partial differentiate equation (3) with respect to $x$ is $f_{zx}=kmx^{k-1}y^{l}z^{m-1}$
Partial differentiate equation (2) with respect to $y$ is $f_{yy}=l(l-1)x^ky^{l-2}z^{m}$
Partial differentiate equation (2) with respect to $z$ is $f_{yz}=lmx^ky^{l-1}z^{m-1}$
Partial differentiate equation (3) with respect to $y$ is $f_{zy}=lmx^ky^{l-1}z^{m-1}$
Partial differentiate equation (3) with respect to $z$ is $f_{zz}=m(m-1)x^ky^{l}z^{m-2}$
