Sin x'in Türevi Nedir ?
Sin x'in türevi cos x'tir.
(sin x)′=cos x
dxd(sin x)=cos x
Sin x'in Türevinin İspatı
1. Yol
f′ (x)=h→0limhf (x+h)−f (x)
(sin x)′=h→0limhsin (x+h)−sin x
sin (p+q)=sin p.cos q+cos p.sin q
(sin x)′=h→0limhsin x.cos h+cos x.sin h−sin x
(sin x)′=h→0limhsin x.cos h−sin x+cos x.sin h
(sin x)′=h→0limhsin x.(cos h−1)+cos x.sin h
(sin x)′=h→0lim [hsin x.(cos h−1)+hcos x.sin h]
(sin x)′=h→0limhsin x.(cos h−1)+h→0limhcos x.sin h
(sin x)′=sin x.h→0limhcos h−1+cos x.h→0limhsin h
t→0limtsin t=1 t→0limtcos t−1=0
(sin x)′=sin x.0+cos x.1
(sin x)′=0+cos x
(sin x)′=cos x
2. Yol
f′ (x)=h→0limhf (x+h)−f (x)
(sin x)′=h→0limhsin (x+h)−sin x
sin p−sin q=2.sin 2p−q.cos 2p+q
(sin x)′=h→0limh2.sin 2x+h−x.cos 2x+h+x
(sin x)′=h→0limh2.sin 2h.cos 22x+h
(sin x)′=h→0limh2.sin 2h.cos 22.(x+2h)
(sin x)′=h→0limh2.sin 2h.cos (x+2h)
(sin x)′=h→0lim21.hsin 2h.cos (x+2h)
(sin x)′=h→0lim2hsin 2h.cos (x+2h)
(sin x)′=h→0lim [2hsin 2h.cos (x+2h)]
(sin x)′=h→0lim2hsin 2h.h→0limcos (x+2h)
h→0 (2h=h)
(sin x)′=h→0limhsin h.h→0limcos (x+h)
(sin x)′=1.cos (x+0)
(sin x)′=1.cos x
(sin x)′=cos x
3. Yol
sin x=x−3!x3+5!x5−7!x7+9!x9−...
cos x=1−2!x2+4!x4−6!x6+8!x8−...
sin x=x−3!x3+5!x5−7!x7+9!x9−...
(sin x)′=(x−3!x3+5!x5−7!x7+9!x9−...)′
(sin x)′=(x)′−(3!x3)′+(5!x5)′−(7!x7)′+(9!x9)′−...
(sin x)′=1−3!3x2+5!5x4−7!7x6+9!9x8−...
(sin x)′=1−3.2!3x2+5.4!5x4−7.6!7x6+9.8!9x8−...
(sin x)′=1−2!x2+4!x4−6!x6+8!x8−...
(sin x)′=cos x
Soru
y=sin 3x⇒y′=dxdy= ?
1. Cevap
y=sin 3x
dy=d(sin 3x)
dy=(sin 3x)′ dx
dxdy=(sin 3x)′
u=3x
du=d(3x)
du=(3x)′ dx
du=3 dx
dxdu=3
y=sin u
dy=d(sin u)
dy=(sin u)′ du
dy=cos u du
dudy=cos u
dxdu.dudy=dxdy
3.cos u=(sin 3x)′
3.cos 3x=(sin 3x)′
2. Cevap
(sin u)′=u′.cos u
(sin 3x)′=(3x)′.cos 3x
(sin 3x)′=3.cos 3x
Published Date:
May 11, 2020
Updated Date:
February 06, 2025