if x^2+y^2=2y.. show that

Question

d^2y/dx^2=1/(1-y)^3

Answer 1

x^2+y^2=2y

Take derivative both sides

2x+2ydy/dx=2dy/dx

x+ydy/dx=dy/dx

(1-y)dy/dx=x

dy/dx=x/1-y

Take derivative both sides

Using d/dx(u/v)=(vu'-uv')/v^2

d^2y/dx^2=((1-y)1-(x)(-dy/dx))/(1-y)^2

                  =((1-y)+xdy/dx)/(1-y)^2

                  =((1-y)+x*x/1-y)/(1-y)^2

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

Where x^2+(1-y)^2=1

So d^2y/dx^2=1/(1-y)^3