Sec x'in Türevi Nedir ?
Sec x'in türevi, tan x.sec x'tir.
dxd(sec x)=tan x.sec x
Sec x'in Türevinin İspatı
1. Yol
f′(x)=h→0limhf(x+h)−f(x)
(sec x)′=h→0limhsec (x+h)−sec x
sec x=cos x1
(sec x)′=h→0limhcos (x+h)1−cos x1
(sec x)′=h→0limhcos x.cos (x+h)cos x−cos (x+h)
(sec x)′=h→0limh.cos x.cos (x+h)cos x−cos (x+h)
(sec x)′=h→0limh.cos x.cos (x+h)−[cos (x+h)−cos x]
cos p−cos q=−2.sin (2p−q).sin (2p+q)
(sec x)′=h→0limh.cos x.cos (x+h)−[−2.sin (2x+h−x).sin (2x+h+x)]
(sec x)′=h→0limh.cos x.cos (x+h)−[−2.sin (2h).sin (22x+h)]
(sec x)′=h→0limh.cos x.cos (x+h)2.sin (2h).sin [22.(x+2h)]
(sec x)′=h→0lim2h.cos x.cos (x+h)sin (2h).sin (x+2h)
(sec x)′=h→0lim[2hsin (2h).cos x.cos (x+h)sin (x+2h)]
(sec x)′=h→0lim2hsin (2h).h→0limcos x.cos (x+h)sin (x+2h)
h=2h(h→0)
(sec x)′=h→0limhsin h.h→0limcos x.cos (x+h)sin (x+h)
t→0limtsin t=1
(sec x)′=1.cos x.cos (x+0)sin (x+0)
(sec x)′=cos x.cos (x+0)sin (x+0)
(sec x)′=cos x.cos xsin x
(sec x)′=cos xsin x.cos x1
tan x=cos xsin x
(sec x)′=tan x.sec x
2. Yol
sec x=cos x1
(sec x)′=(cos x1)′
f(x)=v(x)u(x)⇒f′(x)=[v(x)]2u′(x).v(x)−v′(x).u(x)
(sec x)′=cos2 x(1)′.cos x−(cos x)′.1
(cos x)′=−sin x
(sec x)′=cos2 x0.cos x−(−sin x).1
(sec x)′=cos2 x0−(−sin x)
(sec x)′=cos2 x0+sin x
(sec x)′=cos2 xsin x
(sec x)′=cos x.cos xsin x
(sec x)′=cos xsin x.cos x1
tan x=cos xsin x
(sec x)′=tan x.sec x