第二章一元函数微分学-2导数运算法则

导数的运算四.基本一、和、差、积、商的求导法果函数u(x),vx)在点x处可导,则它们和、差、积、商(分母不为零)在点x处也可导并[u(x)v(x)]u(x)v([u(x)v(x)]u(x)v(x)u(x)v((3)[u(x)]u(x)v(x)u(x)v( (v(x)v( v2( vx

vx

v2x.证(1)、(2)证

设fx)ux),(vx)v(f(x)limf(xx)f(

u(xx)u(limv(x v( limu(xx)v(x)u(x)v(x

v(xx)v( lim[u(xx)u(x)]v(x)u(x)[v(xx)v(

v(xx)v(u(xx)u(x)v(x)u(x)v(xx)v( v(xx)v(u(x)v(x)u(x)v([v(fx)在x处可导推(1)fx+gx+hxfx+gx+hx [fi(x)]i i(2)[u(x)v(x)]u(x)v(x)u(x)v([Cf(x)]Cf(fxgxhxfxgxhxfxgxhx例1求yx32x2sinx的导数 y3x24xcos例 求ysin2xlnx的导数 y2sinxcosxlny2cosxcosxlnx2sinx(sinx)ln2sinxcosxx2cos2xlnx1sin2x例 求ytanx的导数 y(tanx)(sinxcos(sinx)cosxsinx(coscos22cos2x x 2

sec2cos2 cos2(tanx)sec2同理

(cotx)csc2 例4ysecx的导数y(secx)

cosx(cosx) sincos2 cos2

secxtansecxsecxtan同理

(cscx)cscxcot 二、反函数xy)在某区间Iy内单调、可导且y)0,则它的yfx)在对应区间Ix内也可导,且有f(x) (即反函数的导数等于直接函数导数的倒数注：fx0

y0证任取xIx,给x以增量x(x0xxIx由yfx)的单调性可知y于是有y 1

fx)连续 y0(x yf(x)limylim1 x0 y0

(即f(x) (y)例5求函数yarcsinx的导数解xsiny在 (,内单调、可导 1sin21且(siny)cosy 在I1sin21

(siny) cosy

(arcsin(arcsinx)11(arccosx)(arccosx)11记住此公式例 设yarctanx,求解：xtany在区间 ,内单调可导2 2且有tanysec2y1tan2y在对应的区间Ix,内,arctanx (tan

1tan2 111;arccotxarccotx11记住此公例 求函数ylogax的导数解xay在y

()内单调、可导(ay)aylna0,在x

)内有 x)

(ay

ayln

xln 例8yfxx32x的反函数xf1yy2处的导数解：yfx3x2y 2x32x1,x[f1(

ffx(3x2(3x2

15 三、复合函定理如果函数ux)在点x,yf在点ux)可导,则复合yf[x)]在点x可导,且其导数为dydyf(u)(或dydy 或即:函数对自变量求导,等于函数对中间变量求导,注：fu表示fu对变量u求导数，既使量ux fu也是指fu对中间变量u求导数例如f1f11求导x x 1 1fxfx对x求导 它们的关系是 1

11

1 1f

x

f

xx

x

x2

推广设yf(u),uu(v),vvyf{u[vx)]}的导数dydydudv 或ff复合函数的求导法则又称为链式法求yfx在某一点xx0的导数 则fxx fxxxx0fuuu0xx 求函数ylnsinx的导数.解ylnu,usinx.dydydu1cosx cosxcotsin例10

y(x2

的导数 设ux2 ydydydu 10u92x20xx219dy10(x21)9(x2a2x2a2x2

的导数a(a解

(2

)(a2a2x22

ax

a2

xa2a2

2

a2 a2

2

a2 a a2

x11x2aaa2x2 a2x2 a21 a21ax a2x2 a2a2x2 a2 a22a2a2例12yx2x23x

x2)的导数解y1ln(x21)1ln(x y1 2x

x2

3(xsin

x2 3(x例13y

x的导数 sin y

sin )

(1 sin 2 x exex exe解shx

2 exex exechx

shxchx 设ylnx 求lnx,x解ln

lnxx当x0时 y1x当x 时 y 1x1 lnxxlnxx 已知fx可导，求下列函数的导数dy1yfsin2xfcos2x;2yfexef2x.f(sin2x(sin2x)f(cos2x(cos22sinxcosxf(sin2x)2cosxsinxf(cos2sin2x[f(sin2x)f(cos22yf(ex)(ex)ef2(x f(ex)ef2(x)[f2(exf(ex)e

(x)f(ex)e

(x)2f(x)f(ef2(x)[exf(ex)2f(x)f(x)f(ex小结常数和基本初等函数的求导公(C)(sinx)cos(tanx)sec2(secx)secxtan(ax)axln x)

(x)x1(cosx)sinx(cotx)csc2(cscx)cscxcot(ex)ex(lnx) xln (arcsinx)(arctanx)

11x1x

(arccosx)(arccot