If ` x sec theta , y = tan theta ` , then the value of ` (d^(2) y)/(dx^(2)) " at " theta = (pi)/(4)` is

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

C

Explanation

Given , `x = sec theta , y = tan theta`
`(dx)/(dy) = sec theta tan theta , (dy)/(d theta) = sec^(2) theta`
`therefore (dy)/(dx) = (sec^(2) theta)/(sec theta tan theta) = cosec theta`
Now , `(d^(2)y)/(dx^(2))= (d)/(dx) ((dy)/(dx))`
`= (d)/( d theta) (cosec theta) (d theta)/(dx)`
`= - cosec theta cot theta xx (1)/(sec theta tan theta)`
`= -""(1)/(tan^(3) theta)`
At `theta = (pi)/(4) , ((d^(2) y)/(dx^(2)))_( theta = (pi)/(4)) = - ""(1)/((tan ""(pi)/(4))^(3)) = -1`