计算下列函数的二阶导数y''
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/08/23 12:24:36
计算下列函数的二阶导数y''
![](http://img.wesiedu.com/upload/6/79/67998e609cbe557cdc2bde94b76b3c01.jpg)
![](http://img.wesiedu.com/upload/6/79/67998e609cbe557cdc2bde94b76b3c01.jpg)
(1)y'=(e^2x)*(2x)'=2e^2x
y''=2e^2x*(2x)'=4e^2x
(3)y'=sinx+xcosx
y''=cosx+cosx-xsinx=2cosx-xsinx
(5)y'=(x²-1)'/(x²-1)=2x/(x²-1)
y''=[(x²-1)*(2x)'-(2x)*(x²-1)']/(x²-1)²
=(2x²-2-4x²)/(x²-1)²=-2(x²+1)/(x²-1)²
(7)y'=cosx²-xsinx²*(x²)'=cosx²-2x²sinx²
y''=-sinx²*(x²)'-2[sinx²*(x²)'+x²cosx²*(x²)']
= -2xsinx²-4xsinx²-4x³cosx²
(9)x²+xy+y²=3
2x+y+xy'+2yy'=0
y'(x+2y)=-2x-y
y'=-(2x+y)/(x+2y)
y''=-[(x+2y)(2x+y)'-(2x+y)(x+2y)']/(x+2y)²
=-[(x+2y)(2+y')-(2x+y)(1+2y')]/(x+2y)²
(11)(x²-y²)²=4xy
2(x²-y²)(2x-2yy')=4(y+xy')
4x(x²-y²)-4y(x²-y²)y'=4y+4xy'
4x(x²-y²)-4y=y'[4x+4y(x²-y²)]
y'=[x(x²-y²)-y]/[x+y(x²-y²)]
y'=(x³-xy²-y)/(x+x²y-y³)
y''={(x+x²y-y³)[3x²-(y²+2xyy')-y')]-(x³-xy²-y)[1+(2yx+x²y')-3y²]
化简后得y''=6y/(x²+3x³y+9xy³+4x³y^4-4y^6)
(13)y=f(x²)
y'=f'(x²)*(x²)'=2xf'(x²)
y''=2[f'(x²)+xf''(x²)*(x²)']
=2f'(x²)+4x²f''(x²)
不难,主要是计算过程繁复,会乱的
y''=2e^2x*(2x)'=4e^2x
(3)y'=sinx+xcosx
y''=cosx+cosx-xsinx=2cosx-xsinx
(5)y'=(x²-1)'/(x²-1)=2x/(x²-1)
y''=[(x²-1)*(2x)'-(2x)*(x²-1)']/(x²-1)²
=(2x²-2-4x²)/(x²-1)²=-2(x²+1)/(x²-1)²
(7)y'=cosx²-xsinx²*(x²)'=cosx²-2x²sinx²
y''=-sinx²*(x²)'-2[sinx²*(x²)'+x²cosx²*(x²)']
= -2xsinx²-4xsinx²-4x³cosx²
(9)x²+xy+y²=3
2x+y+xy'+2yy'=0
y'(x+2y)=-2x-y
y'=-(2x+y)/(x+2y)
y''=-[(x+2y)(2x+y)'-(2x+y)(x+2y)']/(x+2y)²
=-[(x+2y)(2+y')-(2x+y)(1+2y')]/(x+2y)²
(11)(x²-y²)²=4xy
2(x²-y²)(2x-2yy')=4(y+xy')
4x(x²-y²)-4y(x²-y²)y'=4y+4xy'
4x(x²-y²)-4y=y'[4x+4y(x²-y²)]
y'=[x(x²-y²)-y]/[x+y(x²-y²)]
y'=(x³-xy²-y)/(x+x²y-y³)
y''={(x+x²y-y³)[3x²-(y²+2xyy')-y')]-(x³-xy²-y)[1+(2yx+x²y')-3y²]
化简后得y''=6y/(x²+3x³y+9xy³+4x³y^4-4y^6)
(13)y=f(x²)
y'=f'(x²)*(x²)'=2xf'(x²)
y''=2[f'(x²)+xf''(x²)*(x²)']
=2f'(x²)+4x²f''(x²)
不难,主要是计算过程繁复,会乱的
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