Given: #y=xsqrt(1-x^2) - sin^-1(sqrt(1-x^2))#
Add restrictions to the domain:
#y=xsqrt(1-x^2) - sin^-1(sqrt(1-x^2)); -1<=x<=1#
#dy/dx = (1-2x^2)/sqrt(1-x^2) - 1/sqrt(1-u^2)(du)/dx; u = sqrt(1-x^2), -1 < x < 1#
NOTE: Please observe that the domain must be modified to exclude -1 and +1 to avoid division by 0.
#dy/dx = (1-2x^2)/sqrt(1-x^2) - 1/sqrt(1-(1-x^2))(-x)/sqrt(1-x^2); -1 < x < 1#
#dy/dx = (1-2x^2)/sqrt(1-x^2) + 1/sqrt(x^2)(x)/sqrt(1-x^2);-1 < x < 1#
This is where the proof breaks down; if one simplifies #x/sqrt(x^2) = 1#, then one is discarding the negative half of the domain. The correct action is to write the above as:
#dy/dx = (1-2x^2)/sqrt(1-x^2) + x/sqrt(x^2-x^4);-1 < x < 1#
The proof hits a dead end here
If one allows the assumption that the domain is #0<=x<1#, then #1/sqrt(x^2)# simplifies to become #1/x# (instead of #+-1/x#) and the first derivative becomes:
#dy/dx = (1-2x^2)/sqrt(1-x^2) + 1/x(x)/sqrt(1-x^2);0 <= x < 1#
#x/x# becomes 1:
#dy/dx = (1-2x^2)/sqrt(1-x^2) + 1/sqrt(1-x^2);0 <= x < 1#
Combine like terms:
#dy/dx = (2-2x^2)/sqrt(1-x^2); 0<=x<1#
#dy/dx = 2sqrt(1-x^2); 0<=x<1#
This allows the proof to be completed with a very trivial differentiation:
#(d^2y)/dx^2 = -(2x)/sqrt(1-x^2); 0<=x<1#
However, this assumes the specified domain.
