Answer
$${\mathbf{F}(x, y)=6yz \mathbf{i}+6xz \mathbf{j}+6 xy\mathbf{k}}$$
Work Step by Step
Given
$$f(x, y,z)=6 x y z$$
Since \begin{align}\mathbf{F}(x,y)&=M \mathbf{i}+N\mathbf{j}+P\mathbf{k}\\
&=f_{x}(x, y)\mathbf{i}+f_{y}(x, y)\mathbf{j}+f_{z}(x, y)\mathbf{k}
\end{align}
As, we have
\begin{array}{l}
{f_{x}(x, y)=\frac{\partial f(x,y)}{\partial x}=6yz} \\
{f_{y}(x, y)=\frac{\partial f(x,y)}{\partial y}=6xz } \\
{f_{z}(x, y)=\frac{\partial f(x,y)}{\partial z}=6xy } \\ \end{array}
So, we get
$${\mathbf{F}(x, y)=6yz \mathbf{i}+6xz \mathbf{j}+6 xy\mathbf{k}}$$
