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"If "f(x)= cos x cdot cos 2x cdot cos 4x...

`"If "f(x)= cos x cdot cos 2x cdot cos 4x cdot cos 8x cdot cos 16x," then find "f'((pi)/(4)).`

Text Solution

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`f(x)=(2 sin x cdotcos x cdot cos 2x cdot cos 4x cdot cos 8x cdot cos 16x)/(2 sin x)`
`=(sin 32 x)/(2^(5) sin x)`
`therefore" "f'(x)=(1)/(32)xx(32 cos 32 x xxsin x - cos x xx sin 32 x)/(sin^(2)x)`
`therefore" "f'((pi)/(4))=(32xx(1)/(sqrt(2))-(1)/sqrt(2)xx0)/(32xx((1)/(sqrt(2)))^(2))=sqrt(2)`
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