x3+y3=3y3=3−x3Differentiating implicitly, we get3y2dxdy=−3x2dxdy=−y2x2y=sin(x)tan(x)We take the natural logarithm to obtainlny=tanxlnsinxNext, we take the exponential of both sides to obtainy1⋅dxdy=sec2xlnsinx+tanxsinx1⋅cosx=1+lnsinxsec2x⟹y1⋅dxdy==1+lnsinxsec2xMultiplying both sides by y=sinxtanx, we havedxd(sinxtanx)=(1+lnsinxsec2x)sinxtanxu=2x3+3x2y+xy2+y2The 2nd order partial derivatives is given by uxx,uxy,uyyuxx=12x+6yuxy=6x+2yuyx=6x+2yuyy=2x+2
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