Find the nth differential coefficient of cosx cos2x cos3x
Answers
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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