\f= (18sec3 x - secx)(sec2 x -1) + 6sec5 x - sec3 x =24sec' x - 20sec' x + secx f(5) (x) =120sec' xtanx - 60sec' xtan x + sec.xtan.x (120sec3 x - 60sec' x + secx ) tan x f(6) (x) = (600sec'xtanx - 180sec' xtan x + sec xtan x tan.x + (120sec x - 60sec3 x + secx )sec2 x = (600sec x - 180sec3 x + secx)tan2 x +120sec x -60sec5 x + sec3x =(600sec3 x - 180sec3 x + secx )(sec2 x-1) +120sec x -60sec's x + sec3 x = 720sec x -840sec x +182sec3x-secx (7) (x) = (5,040sec7 x - 4,200sec' x + 546sec' x - secx tan x 7.7 - Taylor & Maclaurin Series | 376 f(8) (x) = (35,280sec7 x - 21,000sec' x + 1,638sec x -secx tan-x + (5, 040sec x - 4,200sec' x + 546sec3 x -secx sec- x =(35,280sec7 x - 21,000sec' x + 1,638sec3 x-secx (sec2 x - 1) + 5,040sec" x - 4,200sec x + 546sec x - sec'x = 40,320 sec x - 60, 480sec x + 23,184sec x -1,640sec x + secx (9) (x) = (362,880sec x - 423,360sec7 x + 115,920sec' x -4,920sec' x + secx tan.x f(10) (x) =(3,265,920sec" x - 2,963,520sec x + 579,600sec' x -14, 760sec'x + secx tan-x= (18sec3 x - secx)(sec2 x -1) + 6sec5 x - sec3 x =24sec' x - 20sec' x + secx f(5) (x) =120sec' xtanx - 60sec' xtan x + sec.xtan.x (120sec3 x - 60sec' x + secx ) tan x f(6) (x) = (600sec'xtanx - 180sec' xtan x + sec xtan x tan.x + (120sec x - 60sec3 x + secx )sec2 x = (600sec x - 180sec3 x + secx)tan2 x +120sec x -60sec5 x + sec3x =(600sec3 x - 180sec3 x + secx )(sec2 x-1) +120sec x -60sec's x + sec3 x = 720sec x -840sec x +182sec3x-secx (7) (x) = (5,040sec7 x - 4,200sec' x + 546sec' x - secx tan x 7.7 - Taylor & Maclaurin Series | 376 f(8) (x) = (35,280sec7 x - 21,000sec' x + 1,638sec x -secx tan-x + (5, 040sec x - 4,200sec' x + 546sec3 x -secx sec- x =(35,280sec7 x - 21,000sec' x + 1,638sec3 x-secx (sec2 x - 1) + 5,040sec" x - 4,200sec x + 546sec x - sec'x = 40,320 sec x - 60, 480sec x + 23,184sec x -1,640sec x + secx (9) (x) = (362,880sec x - 423,360sec7 x + 115,920sec' x -4,920sec' x + secx tan.x f(10) (x) =(3,265,920sec" x - 2,963,520sec x + 579,600sec' x -14, 760sec'x + secx tan-xExercise Find a formula (in Summation Notation if possible) for the given function at x = 0 f (x) = secx Solution f (x) = secx f'(x) = secx tan x f"(x) = secx tan- x + sec' x = secx (sec x -1) + sec3 x = 2 sec'x - secx f(3) ( x) = 6sec' x tan x - sec.xtan.x 6sec' x - secx tan x "(4) (x) = (18sec' xtanx - secrtanx tan x + (6sec' x - secx ) sec- x = (18sec' x - secx )tan x + 6secs x - sec3 x = (18sec3 x - secx (sec2 x -1) + 6sec5 x - sec3 x =24sec' x - 20sec' x + secx f(5) (x) = 120sec' xtanx - 60sec' xtan x + sec xtan.x = (120sec> x - 60sec3 x + secx ) tan x f(6) (x) = (600sec'xtanx -180sec' xtanx + sec xtan x tan x + (120sec x - 60sec' x + secx)sec2 x = (600sec3x - 180sec3 x + secx )tan x +120sec x - 60sec5 x + sec3 x =(600secx - 180sec3 x + secx )sec2x-1)+120sec x -60sec5 x + sec3 x =720sec x -840sec3 x +182sec>x-secx