Question

At the point (1,2) on the curve 4x²+2xy+y²=2 dy/dx = ? and d²y/dx² = ?

Answer 1

Differentiating we get, $8x dx+2y dx +2xdy+2ydy=0$. Hence $(x+y)dy=-(y+4x)dx.$ Hence $dy/dx=-\frac{y+4x}{x+y}.$

Hence $d^2y/dx^2=$ $-\frac{(dy/dx+4)(x+y)-(y+4x)(1+dy/dx)}{(x+y)^2 }$ . Putting the value of $dy/dx$ we get,

$-\frac{3y^2+12x^2+6xy}{(x+y)^3}$ .