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In a recent CM module we've been looking at coupled oscillators and the role of time translational invariance in the description of such physical systems. I will present the statement that I am having trouble understanding and then continue to elaborate.

In stating that a system has a time translational invariance, it follows that

x(t+c) = f(c)x(t)

where x(t) is some function of time, f(c) is some function of proportionality dependent on some constant c, and therefore x(t+c) is the function x at some later time.

After this, it is stated that differentiating with respect to c and setting c=0 gives

d/dt[x(t)] = (omega) x(t) where (omega) = d/dt[ f(c=0) ]

It's then given that x(t) = exp^(omega)(t)

I can clearly see that the exponential form is a solution to differential equation above. My question is how is the differential equation derived with no known form of x(t)? In particular, how does the time derivative of f(c) come into the equation if f(c) is only dependent on c? (I understand that the chain and product rules must be used but wouldn't the time derivative of f(c) return a zero value?)

As well as this, some friends have alluded to Noether's theorem as the governance of this particular rule; is this warranted? I don't see any particular conservation laws here.

Any help or insight is much appreciated!