y=ln4x2+2x+1的几种计算导数方法.docx

y=1n(42+2x+1)的导数计算主要内容：通过对数函数导数公式、导数定义以及函数乘积和函数商的求导法则，介绍y=1n(42+2x+1)的一阶、二阶和三阶导数的主要计算步骤。一阶导数：,对数导数计算Vy=1n(4x2+2x+1),.dy(42+2x+1"8x÷2晟=(4x2+2x+1)=(4x2+2x+1)°.导数定义法计算Vy=1n(4x2+2x+1),.dx,、In4(x+t)2+2(x+t)+1-1n(4x2+2x+1)=Iim(tO),4(x+t)2+2(x+t)+11n4x2+2x÷1=Iim(t0),4x2+2x+1+8xt+4t?+2t1n4x2+2x+1=Iim(t->0)7,z.1.8xt+4t2+2tKim(tT0"*(1+1n42+2+121z/8xt+4t+2t-=Iim(IO)(1÷4+1)tz、r,8xt+4t2÷2t=Iim(1O)(1+In雪而不p42+2x+1)8xt+4t2+2t8xt+"+2t(4x2+2x÷1)tz、-8xt+4t2+2t=1im(t0)(1÷1n42+2x+1)8xt+4t2+2t8x+4t÷24x2+2x÷18x+4t÷2-InJn1代t°)4x，2x+1Ine8x+2"(4x2+2x+1)°二阶导数计算函数商的求导.dy8x+2晟(4x2+2x+1)'.d2y_8(4x2+2x+1)-(8x+2)(8x+2)菽:Z(4x2÷2x÷1)2_32x2+16×+8-64×2-32-4二(4x2÷2x+1)2，-32x216x+84(4x2+2x+1)2，_(32x2+16x-4)=(4x2+2x÷1)2°函数乘积的求导.8x+2*y=(4x2+2x÷1)(4x2+2x+1)y,=8x+2,两边同时对X求导,有:(8x+2)y,+(4x2÷2x+1)y,'=8,将y'代入上式得:,、8x+2z2、(8+2)存诲i+(4+2+i)y''=8,/2、(8x+2)2(4x÷2x÷1)y"8-c42+2+,_8(4x2+2x÷1)-(8x+2)2y"=(4x2÷2x÷1)2，_32x2÷16x-4二一(42+2x+1)2°三阶导数计算：ed2y_32x2+16-4dx2(4x2+2x÷1)2,.业dx3J64x+16)(42+2x+1)2-2(322+16-4)(42+2x+1)(8x+2)二一(4x2+2x+1)4J64x+16)(42+2x+1)-2(322+16-4)(8x+2)=(4x2+2x+1)3'2(32x2+16x-4)(8x÷2)-(32x÷8)(4x2+2x+1)=(4x2+2x÷13)，32(8x3+6x2-3x-1)二(4x2+2x+13)