y=2xarctany/x的二阶导数
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您好，很高兴为您解答
y=2x*arctan(y/x)
y/x=2*arctan(y/x)
u=y/x
u=2*arctanu
两边求解导数
dy/dx=2arctan(y/x)+2x*1/((y/x)^2+1)*(1/x*dy/dx-y/x^2)
=2arctan(y/x)+2x^3*1/(x^2+y^2)*(1/x*dy/dx-y/x^2)
=2arctan(y/x)+2x^2/(x^2+y^2)*dy/dx-2xy/(x^2+y^2)
(1-2x^2/(x^2+y^2))*dy/dx=2arctan(y/x)-2xy/(x^2+y^2)
(y^2-x^2)/(x^2+y^2)*dy/dx=2arctan(y/x)-2xy/(x^2+y^2)
dy/dx=(x^2+y^2)/(y^2-x^2)*[2arctan(y/x)-2xy/(x^2+y^2)
=2(x^2+y^2)/(y^2-x^2)*arctan(y/x)-2xy/(y^2-x^2)
咨询记录 · 回答于2021-11-25
y=2xarctany/x的二阶导数
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您好，很高兴为您解答y=2x*arctan(y/x)y/x=2*arctan(y/x)u=y/xu=2*arctanu两边求解导数dy/dx=2arctan(y/x)+2x*1/((y/x)^2+1)*(1/x*dy/dx-y/x^2)=2arctan(y/x)+2x^3*1/(x^2+y^2)*(1/x*dy/dx-y/x^2)=2arctan(y/x)+2x^2/(x^2+y^2)*dy/dx-2xy/(x^2+y^2)(1-2x^2/(x^2+y^2))*dy/dx=2arctan(y/x)-2xy/(x^2+y^2)(y^2-x^2)/(x^2+y^2)*dy/dx=2arctan(y/x)-2xy/(x^2+y^2)dy/dx=(x^2+y^2)/(y^2-x^2)*[2arctan(y/x)-2xy/(x^2+y^2)=2(x^2+y^2)/(y^2-x^2)*arctan(y/x)-2xy/(y^2-x^2)
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