Differentiate:
f(x)=(secx)/[1+sec(x)]
just plunging in,
f' = ((1+secx)(secx tanx) - secx (secx tanx))/(1+sec^2 x)
= (secx tanx)/(1+secx)^2
or, simplifying things a bi,
f = 1/(1+cosx)
f' = sinx (1+cosx)^2
or, using half-angle formulas,
1+cosx = 2cos^2 (x/2)
f = 1/2 2sec^2 (x/2)
f' = sec(x/2) (sec x/2 tan x/2) (1/2)
= 1/2 sec^3 (x/2) tan(x/2)