Question

find the nth derivative of cosx cos2x cos3x
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Answer 1

Given :  cosx cos2x cos3x

To Find :  nth Derivative

Solution:

y ​= cos2x . cos3x.cosx

cosA. cosB = (1/2)( cos(A + B) + Cos(A - B)

y = cos2x .(1/2)) (cos 4x + Cos2x)

y = (1/2)cos2xcos 4x + (1/2)Cos²2x

y =  (1/2)cos4xcos 2x + (1/2)Cos²2x

1 + cos2A = 2Cos²A

y = (1/2)(1/2)( cos6x + cos2x)  + (1/2)(1/2)(1 + cos4x )

y = 1/4  + (1/4)( cos2x + cos4x  + cos6x)

dⁿcos(ax+ b) = aⁿ cos( ax+ b + nπ/2)

yₙ = nth derivative =  (1/4)  (2ⁿcos(2x +  nπ/2)  + 4ⁿcos(4x +  nπ/2) + 6ⁿcos(6x +  nπ/2))

(1/4)  (2ⁿcos(2x +  nπ/2)  + 4ⁿcos(4x +  nπ/2) + 6ⁿcos(6x +  nπ/2))

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