Is Leibniz rule applied correctly? I wish to know if the Leibniz rule is correctly applied in the following equation or I am missing something:
$$\int \int \frac{\partial f(x,y,z)}{\partial x} dy dz =\frac{\partial  \left(\int \int f(x,y,z)dy dz \right)}{\partial x}  $$ 
$x, y$ and $z$ are independent of each other. The integration limits are constant.
Thanks in advance.

Formally your computation is correct, but to make it rigorous you need to make sure that $f$ satisfies the assumptions of differentiation under the integral sign. 
For the $\textbf {Riemann integral}$ over a bounded set it is sufficient that $f$ is continuously differentiable w.r. to $x$ (i.e. $f$ is continuous, differentiable with respect to $x$ and $\partial_x f$ is also continuous).
In the case of the $\textbf {Lebesgue integral}$, it suffices to assume that
\begin{align*}
&1)\,f(x,\cdot,\cdot)\, \textrm{is integrable for a.e.}\, x,\\
&2)\,\partial_x f(x,y,z)\, \textrm{exists for a.e.}\, x,y,z,\\
&3)\, \partial_x f(x,y,z)\, \textrm{is dominated by an integrable function, i.e., we have} \quad\\
&\quad|\partial_x f(x,y,z)|\le g(y,z)\, \textrm{for a.e.}\,  y,z,\, \textrm{with}\, \int\int |g(y,z)|\,dy\,dz<\infty .   
\end{align*}
