函数导数公式及证明复合函数导数公式)),()0g x ≠'''2)()()()()()()f x g x f x g x g x g x ⎤-=⎥⎦())()x g x ,1．证明幂函数()a f x x =的导数为''1()()a a f x x ax -==证：'00()()()()lim lim n nx x f x x f x x x x f x x x→→+-+-==根据二项式定理展开()nx x +011222110(...)lim n n n n n n n nn n n n n x C x C x x C x x C x x C x x x----→+++++-= 消去0n nn C x x -11222110...lim n n n n n nn n n n x C x x C x x C x x C x x----→++++= 分式上下约去x112211210lim(...)n n n n n n n n n n x C x C x x C x x C x -----→=++++ 因0x →，上式去掉零项111n n n C xnx--==12210()[()()...()]lim n n n n x x x x x x x x x x x x x x----→+-+++++++=12210lim[()()...()]n n n n x x x x x x x x x x ----→=+++++++1221...n n n n x x x x x x ----=++++ 1n n x -=2．证明指数函数()x f x a =的导数为'ln ()x x a a a =证：'00()()()lim lim x x xx x f x x f x a a f x x x+→→+--==0(1)lim x x x a a x→-= 令1xam -=，则有log (1)a x m =-，代入上式00(1)lim limlog (1)x x x x x aa a a mx m →→-==+1000ln ln lim lim lim ln(1)1ln(1)ln(1)ln x x x x x x ma m a a a a m m m a m→→→===+++ 根据e 的定义1lim(1)xx e x→∞=+ ，则10lim(1)m x m e →+=，于是1ln ln limln ln ln(1)x x x x ma a a a a a em →===+3．证明对数函数()log a f x x =的导数为''1()(log )ln a f x x x a==证：'00log ()log ()()()limlim a a x x x x x f x x f x f x x x →→+-+-==00log log (1)ln(1)limlim lim ln aa x x x x x x x x x x x x x a→→→+++===00ln(1)ln(1)lim lim ln ln xxx x x x x xx x x a x a→→++==根据e 的定义1lim(1)xx e x→∞=+ ，则0lim ln(1)x x x x e x →+=，于是 0ln(1)ln 1limln ln ln xxx x e x x ax a x a→+===4．证明正弦函数()sin f x x =的导数为''()(sin )cos f x x x ==证：'00()()sin()sin ()limlim x x f x x f x x x xf x x x→→+-+-== 根据两角和差公式sin()sin cos cos sin x x x x x x +=+00sin()sin sin cos cos sin sin limlim x x x x x x x x x x x x→→+-+-==因0lim(sin cos )sin x x x x →=，约去sin cos sin x x x -，于是0cos sin limx x xx→=因0sin lim1x xx →=，于是sin lim(cos )cos x xxx x→==5．证明余弦函数()cos f x x =的导数为''()(cos )sin f x x x ==-证：'00()()cos()cos ()limlim x x f x x f x x x xf x x x→→+-+-== 根据两角和差公式cos()cos cos sin sin x x x x x x +=-00cos()cos cos cos sin sin cos limlim x x x x x x x x x xx x→→+---==因0lim(cos cos )cos x x x x →=，约去cos cos cos x x x -，于是0sin sin limx x xx→-=因0sin lim1x xx →=，于是sin sin lim()sin x x xx x→=-=-6．证明正切函数()tan f x x =的导数为''21()(tan )cos f x x x==证：'00()()tan()tan ()limlim x x f x x f x x x xf x x x→→+-+-==00sin()sin sin()cos sin cos()cos()cos lim lim cos()cos x x x x xx x x x x x x x xx x x x x →→+-+-++==+根据两角和差公式sin()sin cos cos sin x x x x x x +=+, cos()cos cos sin sin x x x x x x +=-代入上式0(sin cos cos sin )cos sin (cos cos sin sin )limcos()cos x x x x x x x x x x x x x x x→+--=+00cos cos sin (sin sin sin )sin (cos cos sin sin )limlim cos()cos cos()cos x x x x x x x x x x x x x x x x x x x x x→→--+==++ 因22cos sin 1x x +=sin limcos()cos x xx x x x →=+因0sin lim1x xx→=，0lim cos()cos x x x x →+=，上式为20sin 11lim cos()cos cos x x x x x x x →⎡⎤==⎢⎥+⎣⎦7．证明余切函数()cot f x x =的导数为''21()(cot )sin f x x x==-证：'00()()cot()cot ()limlim x x f x x f x x x xf x x x→→+-+-==00cos()cos cos()sin cos sin()sin()sin lim lim sin()sin x x x x xx x x x x x x x xx x x x x →→+-+-++==+根据两角和差公式sin()sin cos cos sin x x x x x x +=+, cos()cos cos sin sin x x x x x x +=-代入上式0(cos cos sin sin )sin cos (sin cos cos sin )limsin()sin x x x x x x x x x x x x x x x→--+=+222200sin sin cos sin sin (sin cos )lim lim sin()sin sin()sin x x x x x x x x x x x x x x x x x→→---+==++ 因22sin cos 1x x +=,且0sin lim1x xx→=，0lim sin()sin x x x x →+=，代入上式20sin 11lim sin()sin sin x x x x x x x →⎡⎤=-=-⎢⎥+⎣⎦8．证明复合函数()()f x g x +的导数为[]'''()()()()f x g x f x g x +=+证:[]'0()()()()()()lim x f x x g x x f x g x f x g x x →+++--⎡⎤+=⎢⎥⎣⎦0()()()()lim x f x x f x g x x g x x x →+-+-⎡⎤=+⎢⎥⎣⎦''()()f x g x =+9．证明复合函数()()f x g x 的导数为[]'''()()()()()()f x g x f x g x f x g x =+证:[]'0()()()()()()lim x f x x g x x f x g x f x g x x →++-⎡⎤=⎢⎥⎣⎦[][]0()()()()()()()()lim x f x x f x f x g x x f x g x x g x x g x x →⎡⎤++-+++-+-=⎢⎥⎣⎦[][]0()()()()()()()()()()lim x f x x f x g x x f x g x x f x g x x f x g x x g x x →⎡⎤+-+++-+++-=⎢⎥⎣⎦[][]0()()()()()()lim x f x x f x g x x f x g x x g x x →⎡⎤+-+++-=⎢⎥⎣⎦0()()()()lim ()()x f x x f x g x x g x g x x f x x x →+-+-⎡⎤=++⎢⎥⎣⎦''()()()()f x g x f x g x =+10．证明复合函数()()f x g x 的导数为'''2()()()()()()()f x f x g x f x g x g x g x ⎡⎤-=⎢⎥⎣⎦证:'0()()()()()lim ()x f x x f x f x g x x g x g x x →+⎡⎤-⎢⎥⎡⎤+=⎢⎥⎢⎥⎢⎥⎣⎦⎢⎥⎣⎦0()()()()lim ()()x f x x g x f x g x x xg x g x x →⎡⎤+-+=⎢⎥+⎣⎦[][]0()()()()()()()()lim ()()x f x x f x f x g x f x g x x g x g x xg x g x x →⎡⎤+-+-+-+=⎢⎥+⎣⎦[][]0()()()()()()()()()()lim ()()x f x x f x g x f x g x f x g x x g x f x g x xg x g x x →⎡⎤+-+-+--=⎢⎥+⎣⎦[][]0()()()()()()lim ()()x f x x f x g x f x g x x g x xg x g x x →⎡⎤+--+-=⎢⎥+⎣⎦[][]0()()()()()()lim ()()x f x x f x g x x g x g x f x x x g x g x x →⎡⎤+-+--⎢⎥=⎢⎥+⎢⎥⎢⎥⎣⎦'''2()()()()()f xg x f x g x g x -= 11．证明复合函数[]()f g x 的导数为[]'''()(())()f g x f g x g x =证:[]'0(())(())(())lim x f g x x f g x f g x x →+-⎡⎤=⎢⎥⎣⎦令()u g x = ,则有()()u g x x g x =+-0())()lim x f u u f u x →+-⎡⎤=⎢⎥⎣⎦ 0())()lim x f u u f u u u x →+-⎡⎤=⎢⎥⎣⎦0())()()()lim x f u u f u g x x g x u x →+-+-⎡⎤=⎢⎥⎣⎦ ''()()f u g x =''(())()f g x g x =12．证明复合函数[]ln ()f x 的导数为[]''()ln ()()f x f x f x =证:令()u f x =,[][]'''ln ()ln f x u u =''1()()f x u u f x == 13．求复合函数x x 的导数解: 令xu x =ln ln u x x =等式左边求导为()''ln u u u=等式右边求导为()'''1ln ln (ln )ln ln 1x x x x x x x x x x=+=+=+ 于是有'ln 1u x u=+,'(ln 1)u x u =+则'()(ln 1)x xx x x =+14. 证明反三角函数arcsin x 的导数为'(arcsin )x =证:令arcsin y x =,则sin y x =对上式两边求导,等式右边'1x =等式左边(根据复合函数求导公式),其导数为''(sin )(cos )y y y = 于是有'(cos )1y y ='1(cos )y y ==再将arcsin y x =代入上式'(arcsin )x ==15. 证明反三角函数arccos x 的导数为'(arccos )x =证:令arccos y x =,则 cos y x =对上式两边求导,等式右边'1x =等式右边(根据复合函数求导公式),其导数为()''cos (sin )y y y =-于是有'(sin )1y y -=,整理后如下： '1(sin )y y =-= 再将arccos y x =代入上式'(arccos )x == 16. 证明反三角函数arctan x 的导数为'21(arctan )1x x =+ 证:令arctan y x =,则 tan y x =对上式两边求导,等式右边'1x =等式右边(根据复合函数求导公式),其导数为()'2'tan (1tan )y y y =+于是有2'(1tan )1y y +=,整理后如下： '211tan y y=+ 再将arctan y x =代入上式'2211(arctan )1tan arctan 1x x x ==++ 17. 证明:反函数的导数为原函数导数的倒数'1''1(),(()0)()f y f x f x -⎡⎤=≠⎣⎦ 如果函数()x y ϕ= 在某区间y I 内单调、可导且'()0y ϕ≠ ，那么它的反函数()y f x =在对应区间 x I 内也可导，并且''1()()f x y ϕ=。