函数导数公式及证明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→→+-+-==V V V V V V根据二项式定理展开()nx x +V011222110(...)lim n n n n n nn nn n n n n x C x C x x C x x C x x C x x x----→+++++-=V V V V V V 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----→++++=V V V V V V 分式上下约去x V112211210lim(...)n n n n n n n n n n x C x C x x C x x C x -----→=++++V V V V 因0x →V ，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----→+-+++++++=V V V V V V 12210lim[()()...()]n n n n x x x x x x x x x x ----→=+++++++V V V V1221...n n n n x x x x x x ----=++++g g1n n x -=g2．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+→→+--==V V V V V V0(1)lim x x x a a x→-=V V V 令1xam -=V ，则有log (1)a x m =-V ，代入上式00(1)lim limlog (1)x x x x x a a a a mx m →→-==+V V V V1000ln 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→→→===+++V V V 根据e 的定义1lim(1)xx e x→∞=+ ，则10lim(1)m x m e →+=V ，于是1ln ln limln ln ln(1)x x x x ma a a a a a em →===+V3．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 →→+-+-==V V V V V V00log log (1)ln(1)limlim lim ln aa x x x x x x x x x x x x x a→→→+++===V V V V V V V V V00ln(1)ln(1)lim lim ln ln xxx x x x xx x x x a x a→→++==V V V V V V根据e 的定义1lim(1)xx e x→∞=+ ，则0lim ln(1)xx x x e x →+=V V V ，于是0ln(1)ln 1lim ln ln ln xxx x e x x a x a x a→+===V V V4．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→→+-+-==V V V V V V 根据两角和差公式sin()sin cos cos sin x x x x x x +=+V V V00sin()sin sin cos cos sin sin limlim x x x x x x x x x x x x→→+-+-==V V V V V V V因0lim(sin cos )sin x x x x →=V V ，约去sin cos sin x x x -V ，于是0cos sin limx x xx→=V V V因0sin lim1x xx →=V V V ，于是sin lim(cos )cos x xxx x→==V V V5．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→→+-+-==V V V V V V 根据两角和差公式cos()cos cos sin sin x x x x x x +=-V V V00cos()cos cos cos sin sin cos limlim x x x x x x x x x x x x→→+---==V V V V V V V因0lim(cos cos )cos x x x x →=V V ，约去cos cos cos x x x -V ，于是0sin sin limx x xx→-=V V V因0sin lim1x xx →=V V V ，于是sin sin lim()sin x x xx x→=-=-V V V6．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→→+-+-==V V V V V V00sin()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 →→+-+-++==+V V V V V V V V V根据两角和差公式sin()sin cos cos sin x x x x x x +=+V V V , cos()cos cos sin sin x x x x x x +=-V V V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→+--=+V V V V V V V00cos 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→→--+==++V V V V V V V V V 因22cos sin 1x x +=0sin limcos()cos x xx x x x →=+V V V V因0sin lim1x xx →=V V V ，0lim cos()cos x x x x →+=V V ，上式为20sin 11lim cos()cos cos x x x x x x x →⎡⎤==⎢⎥+⎣⎦V V V V7．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→→+-+-==V V V V V V00cos()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 →→+-+-++==+V V V V V V V V V根据两角和差公式sin()sin cos cos sin x x x x x x +=+V V V , cos()cos cos sin sin x x x x x x +=-V V V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→--+=+V V V V V V V222200sin 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→→---+==++V V V V V V V V V 因22sin cos 1x x +=,且0sin lim1x xx →=V V V ，0lim sin()sin x x x x →+=V V ，代入上式20sin 11lim sin()sin sin x x x x x x x →⎡⎤=-=-⎢⎥+⎣⎦V V V V8．证明复合函数()()f x g x +的导数为[]'''()()()()f x g x f x g x +=+证:[]'()()()()()()lim x f x x g x x f x g x f x g x x →+++--⎡⎤+=⎢⎥⎣⎦V V V V0()()()()lim x f x x f x g x x g x x x →+-+-⎡⎤=+⎢⎥⎣⎦V V V V V''()()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:[]'()()()()()()lim x f x x g x x f x g x f x g x x →++-⎡⎤=⎢⎥⎣⎦V V V V[][]0()()()()()()()()lim x f x x f x f x g x x f x g x x g x x g x x →⎡⎤++-+++-+-=⎢⎥⎣⎦V V V V V V[][]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 →⎡⎤+-+++-+++-=⎢⎥⎣⎦V V V V V V V [][]0()()()()()()lim x f x x f x g x x f x g x x g x x →⎡⎤+-+++-=⎢⎥⎣⎦V V V V V0()()()()lim ()()x f x x f x g x x g x g x x f x x x →+-+-⎡⎤=++⎢⎥⎣⎦V V V V V V''()()()()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 ⎡⎤-=⎢⎥⎣⎦g g 证:'0()()()()()lim ()x f x x f x f x g x x g x g x x →+⎡⎤-⎢⎥⎡⎤+=⎢⎥⎢⎥⎢⎥⎣⎦⎢⎥⎣⎦V V V V 0()()()()lim ()()x f x x g x f x g x x xg x g x x →⎡⎤+-+=⎢⎥+⎣⎦V V V V V [][]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 →⎡⎤+-+-+-+=⎢⎥+⎣⎦V V V V V[][]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 →⎡⎤+-+-+--=⎢⎥+⎣⎦V V V V V[][]0()()()()()()lim ()()x f x x f x g x f x g x x g x xg x g x x →⎡⎤+--+-=⎢⎥+⎣⎦V V V V V[][]0()()()()()()lim ()()x f x x f x g x x g x g x f x x x g x g x x →⎡⎤+-+--⎢⎥=⎢⎥+⎢⎥⎢⎥⎣⎦V V V V V V'''2()()()()()f xg x f x g x g x -=g g11．证明复合函数[]()f g x 的导数为[]'''()(())()f g x f g x g x =g证:[]'(())(())(())lim x f g x x f g x f g x x →+-⎡⎤=⎢⎥⎣⎦V V V令()u g x = ,则有()()u g x x g x =+-VV0())()lim x f u u f u x →+-⎡⎤=⎢⎥⎣⎦V V V 0())()lim x f u u f u u u x →+-⎡⎤=⎢⎥⎣⎦V V V V V0())()()()lim x f u u f u g x x g x u x →+-+-⎡⎤=⎢⎥⎣⎦V V V V V ''()()f u g x =g''(())()f g x g x =g12．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 =g''1()()f x u u f x ==g 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 xx 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 =g 于是有'(cos )1y y =g'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 =-g于是有'(sin )1y y -=g ,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 =+g于是有2'(1tan )1y y +=g ,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=。