Proving that $\vec F$ is conservative field 
I need to prove that $\vec F$ is coservative field:

$$\vec F=\underbrace{\bigg(yz+\frac{1}{yz} \bigg)}_{Q} \hat i+\underbrace{\bigg(xz-\frac{x}{y^2z} \bigg)}_{P}\hat j+\underbrace{\bigg(xy-\frac{x}{yz^2} \bigg)}_R\hat k$$
My attempt:
$\vec F$ is conservative field iff $\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}=\frac{\partial R}{\partial z}$
but $\frac{\partial Q}{\partial x} \neq \frac{\partial P}{\partial y}$

As James said, you want to find a potential function.  A field $\vec{F}$ is conservative if it can be written as $\vec{\nabla}V$.  In that case you have 
$$ \vec{F} = P \hat{i} + Q \hat{j} + R \hat{k} = \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z}\hat{k}$$
The way you wrote the conservative field condition is wrong for the order of terms.  If defined $P$, $Q$, $R$ as here, the condition is $\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$, and is equivalent to requiring that 
$\tfrac{\partial^2 V}{\partial x \partial y} = \tfrac{\partial^2 V}{\partial y \partial x}$.
