Let t=arcsinxt=arcsinx, which is defined for x in (-pi/2,pi/2)x∈(−π2,π2)
Then:
sint = xsint=x
and
cost = sqrt(1-x^2)cost=√1−x2
because for x in (-pi/2,pi/2)x∈(−π2,π2) the cosine is positive.
So:
y= sin(2arcsinx) = sin2t = 2sintcost = 2xsqrt(1-x^2)y=sin(2arcsinx)=sin2t=2sintcost=2x√1−x2
and using the product rule:
dy/dx = 2xd/dx(sqrt(1-x^2)) +2sqrt(1-x^2)dydx=2xddx(√1−x2)+2√1−x2
dy/dx = 2x(-(2x)/(2sqrt(1-x^2))) +2sqrt(1-x^2)dydx=2x(−2x2√1−x2)+2√1−x2
dy/dx = -(2x^2)/sqrt(1-x^2) +2sqrt(1-x^2)dydx=−2x2√1−x2+2√1−x2
dy/dx = (-2x^2+2-2x^2)/sqrt(1-x^2) dydx=−2x2+2−2x2√1−x2
dy/dx = (2(1-2x^2))/sqrt(1-x^2) dydx=2(1−2x2)√1−x2