`(d^(2)) ((2 cos x cos 3x))/(dx^(2)` is equal to

Options

A. `2^(2)(cos 2x+2^(2)cos 4x)`
B. `2^(2)(cos 2x-2^(2)cos4x)`
C. `2^(2)(-cos 2x+2^(2)cos4x)`
D. `-2^(2)(cos2x+2^(2)cos 4x)`

Answer

D

Explanation

Let `y=2cos x cos 3x`
On differentiating w.r.t. `x` we get
`(dy)/(dx)=2cos x(-3sin 3x)`
`+2cos 3x(-sinx)`
`=-3(sin 4x+sin 2x)_(-1)[sin 4x+sin (-2x)]`
Again differentiaging we get
`(d^(2)y)/(dx^(2))=-3(4cos 4x+2cos 2x)`
`-1(4cos 4x-2cos 2x)`
`=-16 cos 4x-4cos 2x`
`=-4(cos 2x+4cos 4x)`
`=-2^(2)(cos 2x+2^(2)cos 4x)`