Question

Find the nth derivative of cos 5x.cos 3x.cos x.

Answer 1

let y = cos 5x.cos 3x.cos x

`=(cos(5x-3x)+cos(5x+3x))/2cosx{cosA.cosB=1/2[cos(A+B)+cos(A-B)]}`

`=1/2[cos2x.cosx+cos8x.cosx]`

`y=1/4[cos3x+cosx+cos9x+cos7x]`

Take nth derivative,

𝒏𝒕𝒉 𝒅𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆 𝒐𝒇 𝒄𝒐𝒔 (𝒂𝒙+𝒃)=𝒂^𝒏𝒄𝒐𝒔`((npi)/2+ax+b)`

`y_n=1/4[3^ncos((npi)/2+3x)+cos((npi)/2+x)+9^ncos((npi)/2+9x)+7^ncos((npi)/2+7x)]`