第二节导数的运算法则 习题2-2 1.推导余切函数及余割函数的导数公式 csc x (2)(csc x)=-cscxcotx A2()(cot x),cos xy_(cos xy'sinx-(sin x)'cosssircsc2x (2)(cscx)’=( in x)=-cscx x sin x 2.求下列函数的导数 (3) y=2cscx+cotx (4) y=e arccos (7) y=xInxcosx 1+sin t (9)y=xa2(a>0) (2)y'=(n2x)+(2)+(x)(2x)+2hn2+1=-+2ln2+1 (3) y=(2cscx)+(cot x)=-2csc xcot x-cSc (4) y=(e)arccos x +e(arccos arccosr-e arccos (5)y=(x3)log2+x3(og2)=3x2log2+ x(3log2+,) xIn 21 第二节 导数的运算法则 习 题 2-2 1. 推导余切函数及余割函数的导数公式: (1) 2 (cot ) csc x ′ = − x ; (2) (csc ) csc cot x ′ = − x x . 解 (1) 2 2 2 cos (cos ) sin (sin ) cos 1 (cot ) ( ) csc sin sin sin x xx x x x x x x x ′ − ′ ′ ′ = = =− =− . (2) 2 1 1 (csc ) ( ) (sin ) csc cot sin sin x x xx x x ′′ ′ = =− =− . 2. 求下列函数的导数: (1) 3 2 3 y x2 7 x = −+ ; (2) ln 2 2x yx x = + + ; (3) 2csc cot y xx = + ; (4) e arccos x y x = ; (5) 3 2 log x y x = ; (6) ln x y x = ; (7) 2 yx x x = ln cos ; (8) 2 2 1 1 x y x + = − ; (9) ( 0) a x y xa a = > ; (10) 1 sin 1 cos t s t + = + . 解 (1) 3 2 2 3 3 6 yx x (2 ) ( ) (7) 6 x x ′ ′ ′′ = − +=+ . (2) (2 ) 1 (ln 2 ) (2 ) ( ) 2 ln 2 1 2 ln 2 1 2 x xx x yx x x x ′ ′ ′ ′′ = + + = + += + + . (3) 2 y x x xx x ′ ′′ = + =− − (2csc ) (cot ) 2csc cot csc . (4) (e ) arccos e (arccos ) x x y xx ′′ ′ = + 2 2 1 1 e arccos e e (arccos ) 1 1 xxx x x x x =−= − − − . (5) 3 3 23 2 22 2 2 1 1 ( ) log (log ) 3 log (3log ) ln 2 ln 2 xx x x yx x x x x x ′′ ′ = + =+= +