Lekce - Realisticky cz

2( )y′′ = 3x − 4x ′= 6x− 4b)2 xy = x ⋅ e( ) ( ) ( )y′ = x ⋅ e′= x′e + x e′= 2xe + x e2 x 2 x 2 x x 2 x( ) ( )′( ) ( ) ( )y xe x e′x e x e′x′e x e′e xe xe x ex 2 x x x2 2 2 2 x 2 x x x x2 2 22 x′′ = + = + + + = + + + == 2e + 4xe + x ex x 2 x2c) y = log 2x2 1 2 ln 2 2⋅ln 2y′ = ( log2 x′) = ln 22 ( x′) = ⋅ 2x=2x x x⎛ 2⋅ln 2 ⎞′ 2⋅ln 2y′′ = ⎜ ⎟ = −2⎝ x ⎠ xPř. 5:Urči derivace:2 sin xa) ( x e )′b)2⎛ sin x ⎞′⎜ 2 ⎟⎝ x + 1 ⎠c)⎛⎜⎝2x + x + 2x ⎞′⎟⎠a)′ ′ ′( ) ( ) ( )2sin x ( ) ( ) ( )b) ⎜ 2 ⎟+( )c)= + = ⋅ +2 sin x 2 sin x 2 sin x sin x 2 sin xx e x e x e 2x e x e cos x( )( )2 2 2 2 2 2 2sin x′x 1 sin x x 1′⎛ ⎞′ + − + cos x ⋅ 2x x + 1 − sin x ⋅ 2x=22 =22⎝ x 1 ⎠ x + 1 x + 1( )⎛ 2 1 12x x 2x ⎞ ′ ′⎜ + + ⎟ = ⋅ x + x + 2x=⎝ ⎠ 22x + x + 2x1 1 ⎛ 1 1 ⎞2 1 1 ⎛ 2x+ 2 ⎞= ⋅ 1 ( x 2x′⎜ + + ) ⎟ = ⋅ ⎜1+⎟2 2 22222x + x + 2x ⎝ x + 2x ⎠ x + x + 2x⎝ 2 x + 2x⎠Př. 6:Urči derivace:sin x xa) ( e )⋅ ′b)2⎡′⎛ x + 1⎞⎤⎢sin⎜ ⎟⎥⎣ ⎝ 2 + x ⎠⎦c)⎛⎜⎝1( x)2 2sin 2 + x −1⎞′⎟⎠′ ′ ⎡ ′ ′ ⎤⎢⎣⎥⎦sin x⋅x sin x⋅x sin x⋅x sin x⋅xa) ( e ) = e ( sin x ⋅ x) = e ( sin x) ⋅ x + sin x ( x) = e ( cos x ⋅ x + sin x)b)2( + ) − ( + )2 2( x 1) ( 2 x) ( x 1)( 2 x)⎡ ′ ′ ⎤= = =⎣ ⎝ 2 + x ⎠⎦ ⎝ 2 + x ⎠⎝ 2 + x ⎠ ⎝ 2 + x ⎠( 2 + x⎢)⎣⎥⎦2 2 2 2⎡ ⎛ x + 1⎞⎤ ′ ⎛ x + 1⎞⎛ x + 1⎞ ′ ⎛ x + 1⎞+ + − + +sin cos cos⎢ ⎥⎢ ⎜ ⎟⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎢ 2⎥2 2 2 2 2 2⎛ x + 1⎞ 2x 2 x x 1 ⎛ x + 1⎞ 2x + 4x − x − 1 ⎛ x + 1⎞x + 4x−1= cos ⎜ ⎟ = cos cos2 ⎜ ⎟ =2 ⎜ ⎟2⎝ 2 + x ⎠ ⎝ 2 + x ⎠ ⎝ 2 + x ⎠( 2 + x) ( 2 + x) ( 2 + x)3