If y= 1/2 sec^2x prove that (d^y/dx^2)=4y(3y-1) ?

1 Answer
Feb 21, 2018

See below:

Explanation:

#y = 1/2 sec^2x implies#
#dy/dx = 1/2 times 2 sec x times sec x tan x = sec^2x tan x = 2y tanx#
Differentiating once again:
#{d^2y}/{dx^2} =2 dy/dx tanx +2y sec^2x#
But #dy/dx = 2y tan x#. So
#{d^2y}/{dx^2} = 4y tan^2x +2y sec^2x =2 y(2 tan^2 x+sec^2x)#
# = 2y(3 sec^2x-2) = 2y(6y-2) = 4y(3y-1)#