∂u/∂x is simply the derivative of u with all variables except x help constant. Once you get this result, find the partial derivative of the result, this time holding all variables except z constant. That is ∂/∂z(∂u/∂x).
e.g. If you had u = xzy then ∂/∂z(∂u/∂x) = ∂/∂z(zy) = y