#y = 1/2sec^2x#
Use the chain rule to find #dy/dx#
#d/dx[f(x)]^n=n[f(x)]^(n-1)f'(x)#
#d/dx(1/2sec^2x)=1/2 *2secx*secxtanx=sec^2xtanx#
Now use the chain and product rules to find #(d^2y)/dx^2#
#d/dxf(x)g(x)=g(x)f'(x)+f(x)g'(x)#
#d/dxsec^2xtanx=(d^2y)/dx^2 1/2sec^2x=2sec^2xtanxtanx+sec^2xsec^2x=2sec^2xtan^2x+sec^4x=sec^2x(2tan^2x+sec^2x)#
Now use #tan^2x=sec^2x-1# to yield the required result
#sec^2x(2tan^2x+sec^2x)=sec^2x(2(sec^2x-1)+sec^2x)=sec^2x(2sec^2x-2+sec^2x)=sec^2x(3sec^2x-2)#
Now substitute #sec^2x=2y#
#sec^2x(3sec^2x-2)=2y(6y-2)#
#=4y(3y-1)#, as required. #square#
