While reading continuity equation in Fluid mechanics, our professor switched the order of taking time derivative and then applying delta operator to the function. So,I began to think mathematical reason for this.The defination of both operators are:
$$\delta(f(x_i)) := f(x_{i+h})- f(x_i) $$ $$d f(x_i)/dt := \partial f(x_i)/\partial x_i \ \dot x_i + \partial f(x_i)/\partial t$$
Where the last one is due to the chain rule. Now the question is when(All are operators here) :
$$\left[d/dt, \delta\right]= d\delta/dt- \delta d/dt=0 ?$$
I tried to derive the condition of above but I made assumption that $\dot x_i$ is independent of $x_i$. Is this right and are there additional minimalistic conditions for this?
Edit: $x_i$ here is position.