Grade 12th passTrigonometry
Find the nth derivative of cos2x × cos3x ?Please help me !!!!!!!
Find the nth derivative of cos2x × cos3x ?
Please help me !!!!!!!
1 Answer
Arun
Dear student
cos x cos 2x cos 3x = cos 2x [ cos 4x + cos 2x ] /2
=(1/2)[ cos 4x cos 2x + cos^2 2x]
=(1/4) [cos 6x + cos 2x + ( 1 + cos 4x)]
= (1/4) [ 1 + cos 2x + cos 4x + cos 6x ]
Now we have D^n ( cos (ax + b) = a^n cos ( n pi/2 + ax + b).
Thus, D^n( cos x cos 2x cos 3x ) = D^n [ (1/4) { 1 + cos 2x + cos 4x + cos 6x}]
= (1/4) [ 2^n cos(n pi/2 + 2x) + 4^n cos ( n pi/2 + 4x) + 6^n cos( n pi/2 + 6x) ]
=(1/2)[ cos 4x cos 2x + cos^2 2x]
=(1/4) [cos 6x + cos 2x + ( 1 + cos 4x)]
= (1/4) [ 1 + cos 2x + cos 4x + cos 6x ]
Now we have D^n ( cos (ax + b) = a^n cos ( n pi/2 + ax + b).
Thus, D^n( cos x cos 2x cos 3x ) = D^n [ (1/4) { 1 + cos 2x + cos 4x + cos 6x}]
= (1/4) [ 2^n cos(n pi/2 + 2x) + 4^n cos ( n pi/2 + 4x) + 6^n cos( n pi/2 + 6x) ]
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