# Misc 26 - Chapter 13 Class 11 Limits and Derivatives

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

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

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.