高等數(shù)學(xué)上ch25高階導(dǎo)數(shù)

ss(t vdss(t) ddsd2 a

dtdt av(s)定義yfx)fxx處可導(dǎo)， fxxf 稱此極限值為yfxx y，d2y

d2f( f(x).， ，22xy1siny0yyxd2解方程兩邊對x求導(dǎo)：1y1cosyy 2即y

2cosyy1cosyy1siny(y)2 siny

siny

4sin解得

(y)cosy

cosy22cos

(cosyy2sinyy

4sin(2cos

(cosy.xyexey0yyx，求d2解x求導(dǎo)

yxdyexeydy0 x解得dy y

由原方程知x0,y xe ex

且dxx0xe

d2y (exy)(xey)(exy)(1eyy) (xey)2d2d2設(shè)設(shè)xacos3yasin3d2 dy 3asin2tcos tan 3acos2tsind2y

ddy

dtantd

dtant

sec2

3acos2tsin1sec4tcsc(t)并假定(t)，(td2y．y(tyyx)x(t dy(t) (td2 ddy d(t)

d(t)

d(t) dt(t)2

dx(t)

dt(t) ，，(t)(t)(t)(t)

(ty3 d3y，d3f(x)，

f(y，

dny

ddn1y一般地，有y(n)y(n1

dxn1 設(shè)yarctanx,求y y

y( )

21 1 (1x2y

2(1x2

)2(3x2(1x2y(0)y(0)

2x(12x(1x22(3x2(1x2

yeyex，yex，yex，，y(n)e注：yax yaxln yaxln2 ,(ax)(n)axlnnyx（為實數(shù)yx1，y(1)x2，y(n)(1)(n1xn．yxn（n為自然數(shù)）ynxn1，y(n)n(n1)21n!，y(m) (mysinycosxsin(x ycos(x)sin(x2 ，y(n)sin(xn2ysin(kx),yksin(kx，y(n)knsin(kxn ycosysinxcos(x ysin(x)cos(x2 ，y(n)cos(x ycos(k

2y(n)kncos(kxn2yln(1y

x

(x y(1)(x1)2y(1)(2)(x1)3(1)212，(xy(n)

(1)n1(n(x注

x

(1)n1(n(x

即 x

(1)nn!(x

)(n)

anax (ax設(shè)y1a2設(shè)y1a2b2(a0,b0)y(n)

1 (abx)(a

abx

解原式化為y1 a

(1)n(a(1)n

abxa (a y(n)1

n!bn

(1)nn!bn2a(abx) (abx)n1設(shè)ycosaxcosbxy(n解ycosaxcosbx1cos(abxcos(ab 而[cos(ab)x](n)(ab)ncos[(ab)x 2[cos(ab)x](n)(ab)ncos[(ab)xn2y(n)1(ab)ncos[(ab)xn](ab)ncos[(ab)xn2

2yuxvx)，ux)，vxn y(n)C0u(n)vC1u(n1)vC2u(n2)vCnu nnCiu(niv(i （其中u(0)u，v(0)ni設(shè)yx22x3)sinxy(100)y(100)C sin(x100)(x22x3) sin(x99)(2x2)C sin(x98) sinx(x22x3)100cosx(2x2)9900singx)fxxa)2gx)，求f(a)gx)