At the point (1,2) on the curve 4x²+2xy+y²=2 dy/dx = ? and d²y/dx² = ?
Differentiating we get, 8xdx+2ydx+2xdy+2ydy=08x dx+2y dx +2xdy+2ydy=08xdx+2ydx+2xdy+2ydy=0. Hence (x+y)dy=−(y+4x)dx.(x+y)dy=-(y+4x)dx.(x+y)dy=−(y+4x)dx. Hence dy/dx=−y+4xx+y.dy/dx=-\frac{y+4x}{x+y}.dy/dx=−x+yy+4x.
Hence d2y/dx2=d^2y/dx^2=d2y/dx2= −(dy/dx+4)(x+y)−(y+4x)(1+dy/dx)(x+y)2-\frac{(dy/dx+4)(x+y)-(y+4x)(1+dy/dx)}{(x+y)^2 }−(x+y)2(dy/dx+4)(x+y)−(y+4x)(1+dy/dx) . Putting the value of dy/dxdy/dxdy/dx we get,
−3y2+12x2+6xy(x+y)3-\frac{3y^2+12x^2+6xy}{(x+y)^3}−(x+y)33y2+12x2+6xy .
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