If y=tan^2x, show that d^2y/dx^2=2(1+y)(1+3y)

Answer 1

just crank it out
y = tan^2x
y' = 2tanx sec^2x
y" = 2sec^2x sec^2x + 2tanx * 2secx * secx tanx
= 2sec^2x(sec^2x + 2tan^2x)
now, since sec^2x = 1+tan^2x,
= 2(1+y)(1+3y)