函数导数公式及证明doc.docx

1、函数导数公式及证明doc函数导数公式及证明函数类型常量函数幂函数指数函数对数函数三角函数原函数f (x) C ，C 为常量f (x) xaf (x) xmf (x) axf (x) exf ( x) loga xf (x) ln xf (x) sin xf (x) cosx求导公式f ( x) 0( xa ) axa 1( xa )( n) a(a 1).(a n 1)xa n( a 0,1,2., n 1)( xm )( n)m!xm n , (n m)(m n)!( ax )axln a( ax )( n)ax ln n a ,(0 a 1)(ex ) ex(ex )(n )ex(log

2、ax)1x ln a(log ax)(n )( 1)n1 (n 1)! , (0 a 1)xn ln a(ln x)1x(ln x) (n )(1)n 1 (n1)!xn(sin x)cosx(sin x)( n)sin(xn)2(cosx)sin x反三角函数双曲函数反双曲函数f (x) tan xf (x) cot xf (x) arcsinxf (x) arccosxf (x) arctanxf (x) arccot xf ( x) sinh xf ( x) coshxf (x) tanh xf ( x) coth xf (x) arsinh xf (x) arcoshxf (x) ar

3、 tanh x(cosx)( n)cos(xn)2(tan x)sec2 x1x1 (tan x)2cos2(cot x)csc2 x11 (cot x)2sin2x(arcsin x) 11x2(arccos x)11x2(arctan x)11x2(arccot x)11x2(sinh x)coshx(cosh x)sinh x(tanh x)1cosh2 x(coth x)1xsinh2( ar sinh x)1x21( ar cosh x) 1x21(ar tanh x)11x2复合函数导数公式复合函数 求导公式f ( x) g( x)f ( x)gg( x)f (x)g( x) 0g

4、 (x) ,f g (x) f ( x) g(x) f ( x) g ( x) f ( x)gg( x) f (x)gg (x) f ( x)gg ( x)Cgf (x) Cgf (x)f (x)gg (x) f ( x)gg ( x)f ( x)g (x)g 2 (x)f g(x)f ( g( x)gg (x),1证明幂函数 f ( x)xa 的导数为 f ( x)( xa )axa 1证：f (x)lim f ( xVx)f (x)lim( xVx)nxnVx0VxVx0Vx根据二项式定理展开(x)nVxlim (Cn0 xnCn1 xn1 VxCn2 xn 2 Vx2.Cnn 1xVxn

5、1Cnn Vxn ) xnVx0Vx消去 Cn0 xnxnlim Cn1 xn 1VxCn2 xn2Vx2. Cnn 1 xVxn1 Cnn VxnVx0Vx分式上下约去 Vxlim( Cn1 xn 1Cn2 xn2 Vx1 .Cnn 1 xVxn 2Cnn Vxn1)Vx0因 Vx0，上式去掉零项Cn1xn 1nxn1lim ( xVxx)( xVx)n1x( xVx)n 2.xn 2 ( xVx) xn 1Vx0Vxlim( xVx)n 1x( xVx)n 2.xn 2 ( xVx)xn 1Vx0xn 1xgxn2 .xn2 gxxn 1ngxn 12证明指数函数 f ( x)ax 的导数

6、为 (a x )a x ln a证：f (x) lim f ( xVx) f ( x)lima x Vxa xVx 0VxVx0Vxlim ax (aVx1)Vx 0Vx令 aV x 1 m ，则有 Vx loga (m 1) ，代入上式lim ax (aVx1)limaxmVx0VxV x0 log a (m1)limax mlimax ln aax ln aln( m1)0 1lim1Vx0V xln(mVx01)mln am1)ln( m1 )x1根据 e 的定义 elim(1，则 lim( m1)me ，于是xV x0xlimax ln aax ln aaxln a1ln eVx0ln( m1)m3证明对数函数f()logax的导数为f(x)(loga x)1xx ln a证：f (x)limf (xVx)f (x)limloga (xVx)logaxVx0VxVx0Vxlog axVxloga (1Vx)ln(1Vx )limVxxlimxlimxV x 0V x 0VxV x 0 Vx ln ax ln(1Vx )ln(1Vx) VxxlimVxxlimxVx0x ln aV x0x