详细步骤如下:
1.两边取对数，再同时对ⅹ求导，即可得一阶导数，具体步骤如下图所示:
2.本题是求二阶导，先对隐函数两边求导，再用函数商的求导，即可得二阶导数，具体步骤如下:
3.裂项不定积分，步骤如下图所示:
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第1个回答 2020-12-19
第2个回答 2020-12-20
(4). 方程 x^y=y^x能确定函数y=y(x)，求dy/dx;
解：两边取对数得：ylnx=xlny;
两边对x取导数得：y'lnx+(y/x)=lny+(x/y)y'
移项得：[lnx-(x/y)]y'=lny-(y/x)
∴y'=[lny-(y/x)]/[lnx-(x/y)]=(xylnx-y²)/(xylnx-x²)
(5). y=1+xe^y；求d²y/dx²;
解：dy/dx=e^y+x(e^y)(dy/dx)；故dy/dx=(e^y)/(1-xe^y);
d²y/dx²=[(1-xe^y)(e^y)y'-(e^y)(-e^y-xe^y•y')]/(1-xe^y)²
=[(e^y)y'+e^(2y)]/(1-xe^y)²
=[e^(2y)/(1-xe^y)+e^(2y)]/(1-xe^y)²
=[2e^(2y)-xe^(3y)]/(1-xe^y)³;
(6). ∫[1/x²(1+x²)]dx=∫[(1/x²)-1/(1+x²)]dx=∫(1/x²)dx-∫[1/(1+x²)]dx
=-(1/x)-arctanx+C ;
解：两边取对数得：ylnx=xlny;
两边对x取导数得：y'lnx+(y/x)=lny+(x/y)y'
移项得：[lnx-(x/y)]y'=lny-(y/x)
∴y'=[lny-(y/x)]/[lnx-(x/y)]=(xylnx-y²)/(xylnx-x²)
(5). y=1+xe^y；求d²y/dx²;
解：dy/dx=e^y+x(e^y)(dy/dx)；故dy/dx=(e^y)/(1-xe^y);
d²y/dx²=[(1-xe^y)(e^y)y'-(e^y)(-e^y-xe^y•y')]/(1-xe^y)²
=[(e^y)y'+e^(2y)]/(1-xe^y)²
=[e^(2y)/(1-xe^y)+e^(2y)]/(1-xe^y)²
=[2e^(2y)-xe^(3y)]/(1-xe^y)³;
(6). ∫[1/x²(1+x²)]dx=∫[(1/x²)-1/(1+x²)]dx=∫(1/x²)dx-∫[1/(1+x²)]dx
=-(1/x)-arctanx+C ;
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