What is the motivation behind $\tau=\mu \frac{\mathrm{d}u}{\mathrm{d}y}$ for Newtonian fluid? Where does the motivation come from? $\tau$ is the shear stress, $u$ is the velocity and $\mu$ is the shear viscosity. 
EDIT: Since I wrote the question on phone I wasn't clear enough about what I was actually wondering about (see comments below); thus let me clarify. 
I was simply wondering where did the derivative term come from and not why the expression is linear in the derivative term or why it is the first derivative and not e.g. the second and so on. 
The diagram in the answer I accepted shows exactly how the gradient comes into the picture. 
 A: The motivation comes from applying the no-slip boundary condition on a fluid flow. This is probably easier to understand pictorially,

The fluid at the top travels at $u$ while the fluid at the bottom does not move, hence the gradient $\partial u/\partial y>0$. In order to properly model fluid flows, this needs to be accounted for in the Navier-Stokes equations; it is called $\tau$ for "simplifying" the equations.
A: It's a tautology. The viscosity is the proportionality constant for Newtonian fluids. We can imagine a small column of fluid that undergoes a shear strain $\frac{\Delta x}{\Delta y}$ in a time $\Delta t$, so the rate of shear strain is $\frac{\Delta x}{\Delta t \Delta y} \to \frac{\partial u}{\partial y}$ .
In general, one would expect a layer of fluid subjected to some stress (force) to accelerate until it the force acted on it is balanced by the drag force it experiences from the layer underneath it. Then we reach some sort of steady state velocity distribution. From this we would characterize the velocity distribution by some function of stress and velocity:
$\frac{\partial u}{\partial y}=f(\tau, u)$.
Now, in studying fluids like water, one learns that at very low velocities (actually, low Reynolds numbers) that drag force is linear with velocity, i.e., $f_d = - b\, v$. In this case, one could argue that in the (inertial) rest frame of the layer of fluid, it is only the relative velocities of the fluid layers that enter the force equation, so 
$\frac{\partial u}{\partial y}=f(\tau)$.
Furthermore, if the drag force is linear with velocity, i.e., $f_d = - b\, v_\mathrm{rel}$, then the force on a layer of fluid must be equal to the net force acting on it from the layer above or the layer beneath, i.e., $f_d = - b\, v_\mathrm{rel} = -b \frac{\Delta v}{\Delta y} \Delta y$. Note that the mass of the layer is proportional to $\Delta y$, so we would expect it to drop out once we scale our equations appropriately.
I think the point is that Sir Newton wasn't thinking about wet cornstarch when he wrote the equation. For non-Newtonian fluids, the proportionality constant changes with $\frac{du}{dy}$. 
See this chart for example.
