AP Calculus BC - Implicit Differentiation

Question

If xy = x + y², find d²y/dx²

Answer 1

xy' +y = 1+2yy'

(x-2y)y'= 1-y

y' = (1-y)/(x-2y)= 1st derivatiive = dy/dx

take the derivative a 2nd time:

denominator times derivative of numerator minus numerator times derivative of denominator all over denominator squared:

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

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

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

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

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

=(-2x+2xy-2y^2+2)/(x-2y)^3

substitute x+y^2 for xy

=(-2x+2x+2y^2-2y^2+2)/(x-2y)^3

=2/(x-2y)^3

=d²y/dx²

= 2nd derivative

^{no guarantees the calculations are error free, as they get rather tedious}

^{some derivative calculators show y"=(y-1)^2/4xsqrx, y'=sqr(y-1)}

yet I think my solution is more "popular" as well as more likely

Here's another mathematician's answer:

2(1-y)(1-x+3y)/(x-2y)^3

Your instructor or textbook is a tad sadistic

Answer 2

Here is a Desmos graph that actually determines the explicit definitions of the 1st and 2nd derivative.

desmos.com/calculator/ouxfornqwn