Cos 2x'in Türevi Nedir ? cos 2x'in türevi, -2.sin 2x'tir.
d x d ( cos 2 x ) = − 2. s in 2 x
Cos 2x'in Türevinin İspatı 1. Yol f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) ( cos 2 x ) ′ = h → 0 lim h cos 2 ( x + h ) − cos 2 x ( cos 2 x ) ′ = h → 0 lim h cos ( 2 x + 2 h ) − cos 2 x cos ( p + q ) = cos p . cos q − s in p . s in q ( cos 2 x ) ′ = h → 0 lim h cos 2 x . cos 2 h − s in 2 x . s in 2 h − cos 2 x ( cos 2 x ) ′ = h → 0 lim h cos 2 x . cos 2 h − cos 2 x − s in 2 x . s in 2 h ( cos 2 x ) ′ = h → 0 lim h cos 2 x . ( cos 2 h − 1 ) − s in 2 x . s in 2 h ( cos 2 x ) ′ = h → 0 lim 2. h 2. [ cos 2 x . ( cos 2 h − 1 ) − s in 2 x . s in 2 h ] ( cos 2 x ) ′ = h → 0 lim 2 h 2. cos 2 x . ( cos 2 h − 1 ) − 2. s in 2 x . s in 2 h ( cos 2 x ) ′ = h → 0 lim [ 2 h 2. cos 2 x . ( cos 2 h − 1 ) − 2 h 2. s in 2 x . s in 2 h ] ( cos 2 x ) ′ = h → 0 lim 2 h 2. cos 2 x . ( cos 2 h − 1 ) − h → 0 lim 2 h 2. s in 2 x . s in 2 h ( cos 2 x ) ′ = 2. cos 2 x . h → 0 lim 2 h cos 2 h − 1 − 2. s in 2 x . h → 0 lim 2 h s in 2 h h = 2 h ( h → 0 ) ( cos 2 x ) ′ = 2. cos 2 x . h → 0 lim h cos h − 1 − 2. s in 2 x . h → 0 lim h s in h t → 0 l i m t s in t = 1 t → 0 l i m t cos t − 1 = 0
( cos 2 x ) ′ = 2. cos 2 x .0 − 2. s in 2 x .1
( cos 2 x ) ′ = 0 − 2. s in 2 x
( cos 2 x ) ′ = − 2. s in 2 x
2. Yol f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) ( cos 2 x ) ′ = h → 0 lim h cos 2 ( x + h ) − cos 2 x ( cos 2 x ) ′ = h → 0 lim h cos ( 2 x + 2 h ) − cos 2 x cos p − cos q = − 2. s in ( 2 p − q ) . s in ( 2 p + q ) ( cos 2 x ) ′ = h → 0 lim h − 2. s in ( 2 2 x + 2 h − 2 x ) . s in ( 2 2 x + 2 h + 2 x )
( cos 2 x ) ′ = h → 0 lim h − 2. s in ( 2 2 h ) . s in ( 2 4 x + 2 h )
( cos 2 x ) ′ = h → 0 lim h − 2. s in ( 2 2 . h ) . s in [ 2 2 . ( 2 x + h ) ] ( cos 2 x ) ′ = h → 0 lim [ h − 2. s in h . s in ( 2 x + h )] ( cos 2 x ) ′ = h → 0 lim h − 2. s in h . h → 0 lim s in ( 2 x + h ) ( cos 2 x ) ′ = − 2. h → 0 lim h s in h . h → 0 lim s in ( 2 x + h ) t → 0 l i m t s in t = 1
( cos 2 x ) ′ = − 2.1. s in ( 2 x + 0 )
( cos 2 x ) ′ = − 2.1. s in 2 x
( cos 2 x ) ′ = − 2. s in 2 x
3. Yol
sin 2x ve cos 2x fonksiyonlarının sonsuz seri şeklindeki açılımlarından faydalanarak da cos 2x'in türevinin -2.sin 2x'e eşit olduğunu ispatlayabiliriz. Sin 2x ve cos 2x fonksiyonlarının sonsuz seri şeklindeki açılımları aşağıdaki gibidir.
s in 2 x = 2 x − 3 ! ( 2 x ) 3 + 5 ! ( 2 x ) 5 − 7 ! ( 2 x ) 7 + 9 ! ( 2 x ) 9 − ...
cos 2 x = 1 − 2 ! ( 2 x ) 2 + 4 ! ( 2 x ) 4 − 6 ! ( 2 x ) 6 + 8 ! ( 2 x ) 8 − ...
cos 2 x = 1 − 2 ! ( 2 x ) 2 + 4 ! ( 2 x ) 4 − 6 ! ( 2 x ) 6 + 8 ! ( 2 x ) 8 − ...
( cos 2 x ) ′ = [ 1 − 2 ! ( 2 x ) 2 + 4 ! ( 2 x ) 4 − 6 ! ( 2 x ) 6 + 8 ! ( 2 x ) 8 − ... ] ′
( cos 2 x ) ′ = ( 1 ) ′ − [ 2 ! ( 2 x ) 2 ] ′ + [ 4 ! ( 2 x ) 4 ] ′ − [ 6 ! ( 2 x ) 6 ] ′ + [ 8 ! ( 2 x ) 8 ] ′ − ...
( cos 2 x ) ′ = 0 − 2 ! 2.2 x . ( 2 x ) ′ + 4 ! 4. ( 2 x ) 3 . ( 2 x ) ′ − 6 ! 6. ( 2 x ) 5 . ( 2 x ) ′ + 8 ! 8. ( 2 x ) 7 . ( 2 x ) ′ − ...
( cos 2 x ) ′ − 2 ! 2.2 x .2 + 4 ! 4. ( 2 x ) 3 .2 − 6 ! 6. ( 2 x ) 5 .2 + 8 ! 8. ( 2 x ) 7 .2 − ...
( cos 2 x ) ′ = − 2 .1 ! 2 .2.2 x + 4 .3 ! 4 .2. ( 2 x ) 3 − 6 .5 ! 6 .2. ( 2 x ) 5 + 8 .7 ! 8 .2. ( 2 x ) 7 − ...
( cos 2 x ) ′ = − 1 2.2 x + 3 ! 2. ( 2 x ) 3 − 5 ! 2. ( 2 x ) 5 + 7 ! 2. ( 2 x ) 7 − ...
( cos 2 x ) ′ = − 2.2 x + 3 ! 2. ( 2 x ) 3 − 5 ! 2. ( 2 x ) 5 + 7 ! 2. ( 2 x ) 7 − ...
( cos 2 x ) ′ = − 2. [ 2 x − 3 ! ( 2 x ) 3 + 5 ! ( 2 x ) 5 − 7 ! ( 2 x ) 7 + ... ]
( cos 2 x ) ′ = − 2. s in 2 x