From book about hydraulics I saw while reading:
$$ \frac{du_x}{dt} = \frac{\partial u_x}{\partial t} + \frac{\partial u_x}{\partial x} u_x +\frac{\partial u_x}{\partial y} u_y +\frac{\partial u_x}{\partial z} u_z, $$ while $$ \frac{dx}{dt}=u_x\\ \frac{dy}{dt}=u_y \tag{1}\\ \frac{dz}{dt}=u_z $$ How can this be possible? Isn't it be after inserting formula (1): $$ \frac{du_x}{dt} = \frac{\partial u_x}{\partial t} + \frac{\partial u_x}{\partial x} u_x +\frac{\partial u_x}{\partial y} u_y +\frac{\partial u_x}{\partial z} u_z=4\frac{du_x}{dt}? $$
Your first expression is just the total derivative $$ \frac{du_x}{dt} = \frac{\partial u_x}{\partial t} + \frac{\partial u_x}{\partial x} \frac{dx}{dt} +\frac{\partial u_x}{\partial y} \frac{dy}{dt} +\frac{\partial u_x}{\partial z} \frac{dz}{dt}, $$ simplified using the identities (1).
Note that $u_x=u_x(t,x(t),y(t),z(t))$, and $du_x/dt$ is the derivative with respect to all $t$s but $\partial u_x/\partial t$ is only with respect to the first $t$ (derivative with respect to time directly, not via space, so to say).