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Miscellaneous (Term 1 and Term 2)
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Miscellaneous (Term 1 and Term 2)
Last updated at Dec. 8, 2016 by Teachoo
Misc 26 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 4x + 5 sinx3x + 7 cosx Let f (x) = 4𝑥 + 5 sin𝑥3x + 7 cos x Let u = 4x + 5 sin & v = 3x + 7 cos x ∴ f(x) = 𝑢𝑣 So, f’ (x) = 𝑢𝑣′ Using quotient rule f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 Finding u’ & v’ u = 4x + 5sin x u’ = (4x + 5sin x)’ = 4 .1 x1 – 1 + 5 cos x = 4 + 5 cos x & v = 3x + 7 cos x v’ = (3x + 7 cos x)’ = 3 . 1x1 – 1 + 7 ( – sin x) = 3x0 + 7 ( – sin x) = 3 – 7 sin x Now, f’(x) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = (4 + 5 cos𝑥) (3𝑥 + 7 cos𝑥) − (3 − 7 sin𝑥) (4𝑥 + 5 sin𝑥) (3𝑥 + 7 cos𝑥)2 = 4(3𝑥 + 7 cos𝑥)+5 cos𝑥(3𝑥+7 cos𝑥)−3(4𝑥+5 sin𝑥)+7 sin𝑥 (4𝑥+5 sin𝑥) (3𝑥 + 7 cos𝑥)2 = 12𝑥 + 28 cos𝑥 + 15 cos𝑥 + 35 cos2𝑥 −12𝑥 −15 sin𝑥 +28𝑥 sin𝑥 +35𝑠𝑖𝑛2 𝑥 (3𝑥 + 7 cos𝑥)2 = 28 cos𝑥 + 28𝑥 sin𝑥 + 15 𝑥 cos− 15 sin𝑥 + 35 𝑐𝑜𝑠2 𝑥 + 35 𝑠𝑖𝑛2 𝑥 (3𝑥 + 7 cos𝑥)2 = 28( cos𝑥 + 𝑥 sin𝑥) + 15(𝑥 cos𝑥 − sin𝑥) + 35 (𝒔𝒊𝒏𝟐𝒙 + 𝒄𝒐𝒔𝟐 𝒙) (3𝑥 + 7 cos𝑥)2 = 28( cos𝑥 + 𝑥 sin𝑥) + 15(𝑥 cos𝑥 − sin𝑥) + 35 𝟏 (3𝑥 + 7 cos𝑥)2 = 𝟐𝟖( 𝒄𝒐𝒔𝒙 + 𝒙 𝒔𝒊𝒏𝒙) + 𝟏𝟓(𝒙 𝒄𝒐𝒔𝒙 − 𝒔𝒊𝒏𝒙) + 𝟑𝟓 (𝟑𝒙 + 𝟕 𝒄𝒐𝒔𝒙)𝟐