Sin x'in türevi cos x'tir.
Sin x'in Türevi Nedir ? Sin x'in türevi cos x'tir.
( s in x ) ′ = cos x
d x d ( s in x ) = cos x
Sin x'in Türevinin İspatı 1. Yol f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) ( s in x ) ′ = h → 0 lim h s in ( x + h ) − s in x s in ( p + q ) = s in p . cos q + cos p . s in q ( s in x ) ′ = h → 0 lim h s in x . cos h + cos x . s in h − s in x ( s in x ) ′ = h → 0 lim h s in x . cos h − s in x + cos x . s in h ( s in x ) ′ = h → 0 lim h s in x . ( cos h − 1 ) + cos x . s in h ( s in x ) ′ = h → 0 lim [ h s in x . ( cos h − 1 ) + h cos x . s in h ] ( s in x ) ′ = h → 0 lim h s in x . ( cos h − 1 ) + h → 0 lim h cos x . s in h
( s in x ) ′ = s in x . h → 0 lim h cos h − 1 + cos 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
( s in x ) ′ = s in x .0 + cos x .1
( s in x ) ′ = 0 + cos x
( s in x ) ′ = cos x
2. Yol f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) ( s in x ) ′ = h → 0 lim h s in ( x + h ) − s in x s in p − s in q = 2. s in 2 p − q . cos 2 p + q ( s in x ) ′ = h → 0 lim h 2. s in 2 x + h − x . cos 2 x + h + x ( s in x ) ′ = h → 0 lim h 2. s in 2 h . cos 2 2 x + h
( s in x ) ′ = h → 0 lim h 2. s in 2 h . cos 2 2 . ( x + 2 h )
( s in x ) ′ = h → 0 lim h 2. s in 2 h . cos ( x + 2 h )
( s in x ) ′ = h → 0 lim 2 1 . h s in 2 h . cos ( x + 2 h )
( s in x ) ′ = h → 0 lim 2 h s in 2 h . cos ( x + 2 h )
( s in x ) ′ = h → 0 lim [ 2 h s in 2 h . cos ( x + 2 h )]
( s in x ) ′ = h → 0 lim 2 h s in 2 h . h → 0 lim cos ( x + 2 h )
h → 0 ( 2 h = h )
( s in x ) ′ = h → 0 lim h s in h . h → 0 lim cos ( x + h )
( s in x ) ′ = 1. cos ( x + 0 )
( s in x ) ′ = 1. cos x
( s in x ) ′ = cos x
3. Yol s in x = x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 + 9 ! x 9 − ...
cos x = 1 − 2 ! x 2 + 4 ! x 4 − 6 ! x 6 + 8 ! x 8 − ...
s in x = x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 + 9 ! x 9 − ...
( s in x ) ′ = ( x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 + 9 ! x 9 − ... ) ′
( s in x ) ′ = ( x ) ′ − ( 3 ! x 3 ) ′ + ( 5 ! x 5 ) ′ − ( 7 ! x 7 ) ′ + ( 9 ! x 9 ) ′ − ...
( s in x ) ′ = 1 − 3 ! 3 x 2 + 5 ! 5 x 4 − 7 ! 7 x 6 + 9 ! 9 x 8 − ...
( s in x ) ′ = 1 − 3 .2 ! 3 x 2 + 5 .4 ! 5 x 4 − 7 .6 ! 7 x 6 + 9 .8 ! 9 x 8 − ...
( s in x ) ′ = 1 − 2 ! x 2 + 4 ! x 4 − 6 ! x 6 + 8 ! x 8 − ...
( s in x ) ′ = cos x
4. Yol e i x = cos x + i . s in x
( cos x + i . s in x ) ′ = ( e i x ) ′
( cos x ) ′ + ( i . s in x ) ′ = ( e i x ) ′
( cos x ) ′ + i . ( s in x ) ′ = ( e i x ) ′
( e u ) ′ = u ′ . e u
( cos x ) ′ + i . ( s in x ) ′ = ( i x ) ′ . e i x
( cos x ) ′ + i . ( s in x ) ′ = i . e i x
( cos x ) ′ + i . ( s in x ) ′ = i . ( cos x + i . s in x )
( cos x ) ′ + i . ( s in x ) ′ = i . cos x + i 2 . s in x
i 2 = − 1
( cos x ) ′ + i . ( s in x ) ′ = i . cos x + ( − 1 ) . s in x
( cos x ) ′ + i . ( s in x ) ′ = i . cos x + ( − s in x )
( cos x ) ′ + i . ( s in x ) ′ = − s in x + i . cos x
( s in x ) ′ = cos x