Derivatives by formula - sin & cos

Chapter 12 Class 11 Limits and Derivatives
Concept wise

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### Transcript

Misc 20 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): a + b sin﷮x﷯﷮c + d cos x﷯ Let f (x) = a + b sin﷮x﷯﷮c + d cos x﷯ Let u = a + b sin x & v = c + d cos x So, f(x) = 𝑢﷮𝑣﷯﷯ ∴ f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = a + b sin x u’ = (a + b sin x)’ = 0 + b cos x = b cos x & v = c + d cos x v’= 0 + d (– sin x) = – d sin x f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 𝑏 cos﷮𝑥 (𝑐 + 𝑑 cos﷮𝑥) − (− 𝑑 sin﷮𝑥) (𝑎 + 𝑏 sin﷮𝑥)﷯﷯﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 𝑐𝑜𝑠2𝑥 + (𝑑 sin﷮𝑥) (𝑎 + 𝑏 sin﷮𝑥)﷯﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 𝑐𝑜𝑠2𝑥 + 𝑎𝑑 sin﷮𝑥 +𝑏𝑑 sin2﷮𝑥﷯﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 (𝒄𝒐𝒔𝟐𝒙 + 𝐬𝐢𝐧𝟐﷮𝒙﷯) + 𝑎𝑏 sin﷮𝑥﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 𝟏﷯+ 𝑎𝑑 sin﷮𝑥﷯ ﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑎𝑑 sin﷮𝑥 ﷯+ 𝑑𝑏 ﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ ∴ f’(x) = 𝒄𝒃 𝒄𝒐𝒔﷮𝒙 + 𝒂𝒅 𝒔𝒊𝒏﷮𝒙 ﷯+ 𝒅𝒃 ﷯﷮ (𝒄 + 𝒅 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.