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Derivatives by formula - sin & cos
Last updated at May 29, 2023 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