高數(shù)課件2.函數(shù)的微分

一、微分的定引例邊長(zhǎng)由x0到x0x, 設(shè)薄片邊長(zhǎng)為x，面積為A，則Ax2 當(dāng)x在0 )A0A(xx)2 02xx0 A

2 定義yf(x在點(diǎn)x0yf(x0x)f(x0

yAxo(x)A為不依賴于△xyf(xx0Axyf(xx0自變量的改變量△x的微分，記作dy或df dydy是△y的說(shuō)明yf(x在點(diǎn)x0①、yf(xx0處的微分dy是自變量的x的線性函數(shù)；② ydyo(x)即ydy是比自變量的改變量△x③、A0時(shí)，dy與△yydyo(x)1o(x)1dy dy Ax

(x二、函數(shù)可微的條定理yf(xx0處可微的充要條件y=f(x)在點(diǎn)x0處可導(dǎo)，且Af(x0) dyf(x0（必要性）yf(xx0yf(x0x)f(x0)Ax limy

o(x)x0

lim

x0 yf(xx0Af(x0yf(xx0limyf(xx0 yf(x) (lim 故

yf(x0)xxf(x0)x即dyf(x0 求函數(shù)yx2在x=1和x=3處的微分解yx2x1處的微分為dy(x2x1x2xx3dy(x2)x3x 求函數(shù)yx3當(dāng)x2,x0.02時(shí)的微分解dyx3x3x2x,

x2x0.02dy

3220.02三、微分的幾何意 dyf(x0)xtanx dyf(x)dx

yf dyf(x)

x0四、基本初等函數(shù)的微 與微分運(yùn)算法 （Pu(xv(x均可微，則d(uvdu d(CuCdu(C為常數(shù)d(uv)vdu duvduudv(vvv vv d(uv)(uv)dx(uvuv)duvdxuvdxvduud 求函數(shù)yx3e2x的微分解dyd(x3e2x)e2xd(x3x3d(e2xe2x3x2dxx32e2xdxx2e2x(32x)dx 求函數(shù)ysinx的微分xdydsinxxd(sinx)sin xcosxdxsinxdx xcosxsinxdx yf(uu(x)yf(x)dyyxdxf(u)(x)ddyf說(shuō)明這種特性稱為特點(diǎn)：uyfdyf 求函數(shù)ysin(2x1)的微分解令ysinu,u2x1 dyd(sinu)cos(2x1)2dx2cos(2x222 yln(1ex),求d22 dy

212

d(1ex)

212

d(x2 1

ex2xdx

212 yln(x x21) 求dxx2xx2

d(x x21)

dx

x2xx2xx2x2 ye13xcosx,求d dyd(e13xcosx)cosxd(e13x)e13xd(coscosxe13x(3dx)e13xd(sine13x(3cosxsin 設(shè)ysinxcos(xy)0 求d解xdysinxycosxsin(x

1

dy

dyycosxsin(xy)dxsin(xy)sinx思考如何用一階微分形式不變性提示sinxdyyd(sinxsin(xy)(dxdy)①、

222

)d(1sint