2-3高等数学同济大学第六版本

习题习题231求函数的二阶导数(1)y2x2lnx(2)ye2x1yxcosxyetsint(5)y a2x2(6)yln(1(6)yln(1x2)(7)ytanx1(8)yx31(9)y(1x2)arctanx(10)yexx(11)yxex2(12)yln(x1x2)解(1)y4x1xy41(2)ye2x122e2x1x2y2e2x124e2x1yxcosxycosxxsinxysinxsinxxcosx2sinxxcosxyetsintetcostet(costsint)yet(costsint)et(sintcost)2etcost(5)y12 a2x2(a2x2)xa2x2ya2x2xa2x2xa2x2a2(a2x2) a2x2(6) y11(1x2)2x1yyx2)2x(2x)x2)x2)2x2)2(7)ysec2x(7)ysec2xy2secx(secx)2secxsecxtanx2sec2xtanx(8)y(x3(x31)23x2(x3y6x(x33x22(x36x(2x3(x31)4(x3(9)y2xarctanx(1x2)112xarctanx1y2arct2x 1x2(10)yexxex1ex(xx2x2x4(11)ex2xex22)ex22x2)y[ex(xex]x2ex(x1)2xex(x22xx3yex22x2x2)ex24x2xex232x2)(12)y1x 1x2(x 1x2)1x 1x2(12x21x2)11x2y11(1x2)112x21x2)xx)2 12设f(x)(x10)6 f(2)?解f(x)6(x10)5 f(x)30(x10)4 f(x)120(x10)3f(2)120(210)32073603f(x)yd2ydx2(1)yf(x2)(2)yln[f(x)]解(1)yf(x2)(x2)2xf(x2)(2)(2)yy2f(x2)2x2xf(x2)2f(x2)4x2f(x2)1f(x)f(x)yff(x)ffff(x)[f[f(x)]2[f(x)]24试从dx4试从dx1导出dyy(1)d2xdy2(y)3y(2)d3xdy3(y)5解(1)d2x ddxdy2dydyd1dyyd1dxdxy dy(y)2 1yy(y)3(2)d3xdydy3dydydxdxdyyy3(y)2y13(y(y)6y(y)555是常数并验证d2s2s0dt2dtd2sA2sintdt2d2s就是物体运动的加速度dt2d2s2sA2sit2Asit0dt26yC1exC2ex(C1C2)满足关系式解dsAcosty2y01 解yC1 yC2exC

2ex1

22exC2ex)2(CexC

ex)1 2 1 21 2 1 (C2exC2ex)(C2exC2ex)01 2 1 7yexsinx满足关系式y2y2y0解yexsinxexcosxex(sinxcosx)yex(sinxcosx)ex(cosxsinx)2excosxy2y2y2excosx2ex(sinxcosx)2exsinx2excosx2exsinx2excosx2exsinx08求下列函数的n阶导数的一般表达式1 2 n1 n 1 2 (1)yxnaxn1axn2a xa(aa1 2 n1 n 1 2 ysin2xyxlnxyxex1 2 解(1)ynxn1(n1)axn2(n2)axn31 2 1 2 yn(n1)xn2(n1)(n2)axn3(n2)(n3)axn41 2 y(n)n(n1)(n2)21x0n!(2)y2sinxcosxsin2xy2cosx2six)2y22co)22si2)2 2 y(4)co22)23si32)y(n)2n1si(n]2(3) ylnx1y1x1xy(1)x2y(4)(1)(2)x3y(n)(1)(2)(3)(n2)xn1(1)n2(n2)!(1)n(nxn1 xn1(4) yexxexyexexxex2exxexy2exexxex3exxexy(n)nexxexex(nx)9求下列函数所指定的阶的导数yexcosxy(4)(2)yxshx求y(100)(3)yx2sin2x求y(50).解(1)令uexvcosx有uuuu(4)exvsinxvcosxvsinv(4)cosx所以 y(4)u(4)v4uv6uv4uvuv(4)ex[cosx4(sinx)6(cosx)4sinxcosx]4excosxuxvsh则有u1u0vchxvshxv(99)chxv(100)shx所以u(100)vC1

u(99)vC2

u(98)vC98uv(98)C99uv(99)uv(100)100

100

100

100100chxxshx(3)令ux2vsin2x则有u2xu2u0v(48)248sin(2x48)248si2C48C48uv(48)C49uv(49)uv(50)505050492228sin2x502x