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Misc 26 - Find derivative: 4x + 5 sin x / 3x + 7 cos x - Derivatives by formula - sin & cos

Misc 26 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Misc 26 - Chapter 13 Class 11 Limits and Derivatives - Part 3


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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 sin﷮x﷯﷮3x + 7 cos﷮x﷯﷯ Let f (x) = 4𝑥 + 5 sin﷮𝑥﷯﷮3x + 7 cos x﷯ Let u = 4x + 5 sin & v = 3x + 7 cos x ∴ f(x) = 𝑢﷮𝑣﷯ So, f’ (x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ 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) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = (4 + 5 cos﷮𝑥)﷯ (3𝑥 + 7 cos﷮𝑥)﷯ − (3 − 7 sin﷮𝑥)﷯ (4𝑥 + 5 sin﷮𝑥)﷯﷮ (3𝑥 + 7 cos﷮𝑥﷯)﷮2﷯﷯ = 4(3𝑥 + 7 cos﷮𝑥)+5 cos﷮𝑥(3𝑥+7 cos﷮𝑥)−3(4𝑥+5 sin﷮𝑥)+7 sin﷮𝑥 (4𝑥+5 sin﷮𝑥)﷯﷯﷯﷯﷯﷯﷮ (3𝑥 + 7 cos﷮𝑥﷯)﷮2﷯﷯ = 12𝑥 + 28 cos﷮𝑥 + 15 cos﷮𝑥 + 35 cos2﷮𝑥 −12𝑥 −15 sin﷮𝑥 ﷯+28𝑥 sin﷮𝑥 +35𝑠𝑖𝑛2 𝑥 ﷯ ﷯﷯﷯﷮ (3𝑥 + 7 cos﷮𝑥﷯)﷮2﷯﷯ = 28 cos﷮𝑥 + 28𝑥 sin﷮𝑥 + 15 𝑥 cos﷮− 15 sin﷮𝑥 + 35 𝑐𝑜𝑠2 𝑥 + 35 𝑠𝑖𝑛2 𝑥﷯﷯﷯﷯﷮ (3𝑥 + 7 cos﷮𝑥﷯)﷮2﷯﷯ = 28( cos﷮𝑥 + 𝑥 sin﷮𝑥) + 15(𝑥 cos﷮𝑥 − sin﷮𝑥) + 35 (𝒔𝒊𝒏𝟐𝒙 + 𝒄𝒐𝒔𝟐 𝒙)﷯﷯﷯﷯﷮ (3𝑥 + 7 cos﷮𝑥﷯)﷮2﷯﷯ = 28( cos﷮𝑥 + 𝑥 sin﷮𝑥) + 15(𝑥 cos﷮𝑥 − sin﷮𝑥) + 35 𝟏﷯ ﷯﷯﷯﷯﷮ (3𝑥 + 7 cos﷮𝑥﷯)﷮2﷯﷯ = 𝟐𝟖( 𝒄𝒐𝒔﷮𝒙 + 𝒙 𝒔𝒊𝒏﷮𝒙) + 𝟏𝟓(𝒙 𝒄𝒐𝒔﷮𝒙 − 𝒔𝒊𝒏﷮𝒙) + 𝟑𝟓 ﷯﷯﷯﷯﷮ (𝟑𝒙 + 𝟕 𝒄𝒐𝒔﷮𝒙﷯)﷮𝟐﷯﷯

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.