[tex]\nabla f(x,y,z)=\mathbf f(x,y,z)=yze^{xz}\,\mathbf i+e^{xz}\,\mathbf j+xye^{xz}\,\mathbf k[/tex]
[tex]\dfrac{\partial f(x,y,z)}{\partial x}=yze^{xz}[/tex]
[tex]\implies f(x,y,z)=\displaystyle\int yze^{xz}\,\mathrm dx=\dfrac{yz}ze^{xz}+g(y,z)[/tex]
[tex]f(x,y,z)=ye^{xz}+g(y,z)[/tex]
[tex]\dfrac{\partial f(x,y,z)}{\partial y}=e^{xz}=e^{xz}\dfrac{\partial g(y,z)}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g(y,z)}{\partial y}=0\implies g(y,z)=h(z)[/tex]
[tex]f(x,y,z)=ye^{xz}+h(z)[/tex]
[tex]\dfrac{\partial f(x,y,z)}{\partial z}=xye^{xz}=xye^{xz}+\dfrac{\partial h(z)}{\partial z}[/tex]
[tex]\implies\dfrac{\partial h(z)}{\partial z}=0\implies h(z)=C[/tex]
[tex]f(x,y,z)=ye^{xz}+C[/tex]
