高數(shù)課件-2隱函數(shù)2.4和參數(shù)方程求導(dǎo)

一、隱函一、隱函數(shù)的導(dǎo)若函數(shù)y可以由x的解析表達(dá)式直接表示，稱類函數(shù)為顯函數(shù). 例如，yx21.F(x,y)這類函數(shù)為隱函數(shù). 例如，lnysin(xy).31例如，xy310可以化成31

F(x,y)dF(x,y)0dx說(shuō)明求隱函數(shù)的導(dǎo)數(shù)時(shí)，只需將確定隱函數(shù)的方程兩邊x求導(dǎo)，凡遇到含有y的yyxd 求由方程eyxye0所確定的隱函數(shù)d 解x d(eyxye)0 eydyyxdy

dy xey

(xey例 求由方程y52yx3x70確定的隱函yy(xx0解x

x0

5y4dy2dy121x60dx dxdy121x6 5y42, x0y0

x0例 求橢圓x2y2

1點(diǎn)2, 處的切線方程 解xx2yy dy9x

x2y

k33 16 33

y

(x333 3333 3x4y 3例 求由方程xy1siny0所確定的隱函2d2 dx2解x1dy1cosydy dy x2sinydd2ydx2

dx(2cos

4sin(2cos 設(shè)y=y(x)由方程eyxye確定，求y(0),解x

eyyyxy eyy2(eyx)y2y x0時(shí)，y1y(0)e

y(0)二、對(duì)數(shù)求導(dǎo)這種方法稱為.yu(x)vx)(u(x)0u(x)不恒等于1lnyv(x)lnu(x)yv(x)lnu(x)v(x)u(x) u(x)yu(x)v(x)v(x)lnu(x)v(x)u(x) u(x) 求yxsinx(x0)的導(dǎo)數(shù)解lnysinxlnx1ycosxlnxsinx yxsinx(cosxlnxsinxx 求(cosy)x(sinx)y的導(dǎo)數(shù)解xlncosyylnsinxlncosyx (siny)yylnsinxycoscos sinylncosyycot xtanylnsin說(shuō)明對(duì)數(shù)求導(dǎo)法還常 axbax 求y (a0,b0 1) 解lnyxlnaalnblnxblnxlnax

ylnaab

axbaxb b ln 求y

(x1)(x(x(x1)(x(x3)(xlny1ln(x1)ln(x2)ln(x3)ln(x4)2x y1 2x x x x4y

(x1)(x(x3)(x x(x1)(x(x3)(xx12x3的情形可類似求解三、參數(shù)方程表示的函數(shù)的x(t)yx(tt1(x 假設(shè)(t),(t)可導(dǎo)，且(t)0 dydtdy

d

d d

dxdxdtdx d (t),(t二階可導(dǎo)，且(t0yf(x可求二階導(dǎo)數(shù)

dy(t) d2yddyddy d dxdx dtdx d(t)(t)(t)2

(t)(t)(t)(t)3

d2 (t)注意

dx(t)

dx2(t)例 yy(x的二階導(dǎo)數(shù)

xyln(1t2

d1d11t1t

ddy d2 ddy

dx

1

2(1t2d dxdx

dx

1txa(tsin例 求由擺線的參數(shù)方程ya(1cost)所表yy(x的二階導(dǎo)數(shù)d解dy asind aaddy

1d2 ddy

dx d2 dxdx

dx

a(1cost)2)(t2n,n)例

xt2

(0t

ysinyyy(x的導(dǎo)數(shù)解tdx2t2dt2tdycosydy0dt dt

dx2(t1)dtdy d 1cos dy

d(t1)(1d例

ey

x3t2sinty10

.dddd解tdy eycostdt 1eysintdddt

d d dt

t

ey(1ey(1eysint)(6t四、極坐標(biāo)表示的曲線的斜rr()xrcos,yrsinxr()cosyr(則dy

r()sinr()dr()cosd例 求心形線ra(1cos)在處的斜率2解xr()cosa(1cos)cosyr()sina(1cos

dy

ddd2dx2

coscossinsinxx(tyy(tx d d①、找出相關(guān)變③、求出未知的相關(guān)變化率 其速率為140mmin, 當(dāng)氣球高度為500m時(shí)，觀察員解t分后其