I came across these equations in a mechanics textbook and wish to differentiate the first equation w.r.t. time to obtain that second equation. Any help is much appreciated!
$c(f_1+f_2)=a^2(\frac{1}{2}\dot{a}^2 -ca+h)+2af'_2$
$c(1-\frac{\dot{a}}{c})(f'_1+f'_2)=ca(-2\dot{a}^2 (1-\frac{1}{2}\frac{\dot{a}}{c})-a\ddot{a}(1-\frac{\dot{a}}{c})+2\frac{\dot{a}}{c}h+\frac{a}{c}\dot{h})+2a(1+\frac{\dot{a}}{c}){f''_2}$
Here: $f_1(x)=f_1(t-r/c)$ and $f_2(x)=f_1(t+r/c)$
The book didn't explicitly state the functions of time, but from the second equation we can see that $a$ and $h$ are all functions of time.
We also know that $\frac{f'_1+f'_2}{a}+\frac{1}{2}\dot{a}^2+h=0$ (if that's of any help)
I tried doing it a few times and this is all I get:
$$c(\dot{f_1}+\dot{f_2})=a\dot{a}^2+a^2\dot{a}-3ca^2+\dot{h}a^2+2ha+2\dot{a}{f'_2}+2a\dot{f'_2}$$
Please let me know where I'm going wrong. Thanks