Find the first and second derivative of any function defined implicity yx2+y2=x

Question

Find the first and second derivative of any function defined implicity by yx²+y²=x

Answer 1

x^2 y + y^2 = x

first derivative:

x^2 * y' + 2x y + 2y y' = 1

x^2 * y' + 2y y' + 2xy = 1

y' (x^2 + 2y) + 2xy = 1

y' = (1 - 2xy)/(x^2+2y)

2nd derivative:

y'' = [(x^2+ 2y)(-2) (xy' + y) - (1-2xy)(2)(x+y')]/(x^2+2y)^2

= [ (-2) (x^2+2y)(xy' + y) + (1-2xy)(x+y')]/(x^2+2y)^2

where y' is given above in terms of x and y