How do you differentiate # f(x)=(5x^6 - 4) sin(6x) sin(3x) cos(5x) # using the product rule?

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Answer 1 · 2015-12-20T16:53:28

f '(x) = #30x^5(1/4)[cos 8x -cos14x +cos 2x- cos 4x]#

#+(5x^6 -4) (14 sin 14x -8sin8x +4 sin4x -2sin 2x)#

Product rule would be more convenient if trignometrical functions are simplified first. Accordingly,

sin 6x sin 3x cos 5x = (sin 6x cos 5x) sin 3x

=#1/2#(sin 11x + sin x) sin 3x

=#1/2#[sin 11x sin3x + sin x sin 3x]

= #1/4#[cos 8x -cos14x +cos 2x- cos 4x]

Now f '(x) = #30x^5(1/4)[cos 8x -cos14x +cos 2x- cos 4x]#

#+(5x^6 -4) (14 sin 14x -8sin8x +4 sin4x -2sin 2x)#