Electrical Engineering Math, showing time independence? how?
\$ T_c(t)\$ is indepent from time, because the phase difference does not depend on the moment it is measured.

You already know that $$\Delta\phi(t) = 2\pi f\frac{1}{c}(2vt-2d)$$

Putting \$t+\Delta t\$ in place of \$t\$ in that formula gives you $$\Delta\phi(t+\Delta t)=2\pi f\frac{1}{c}(2v(t+\Delta t)-2d)$$

Now you can calculate the difference: $$\Delta\phi(t+\Delta t)-\Delta\phi(t)=2\pi f\frac{1}{c}(2v(t+\Delta t)-2d)-2\pi f\frac{1}{c}(2vt-2d)=2\pi f\frac{1}{c}(2v(t+\Delta t-t)-2d+2d)=2\pi f\frac{2v}{c}\Delta t$$
, which is the part you were asking for, if I'm not mistaken.