Ex 5.5, 8 - Differentiate (sin x)x + sin-1 root x - Chapter 5 - Ex 5.5

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.5, 8 Differentiate the functions in, ( sin﷮𝑥﷯)﷮𝑥﷯+ sin﷮−1﷯ ﷮𝑥﷯ Let 𝑦= sin﷮𝑥﷯﷯﷮𝑥﷯ + sin﷮−1﷯﷮ ﷮𝑥﷯﷯ Let 𝑢 = sin﷮𝑥﷯﷯﷮𝑥﷯ & 𝑣 = sin﷮−1﷯﷮ ﷮𝑥﷯﷯ 𝑦 = 𝑢 + 𝑣 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 (𝑢 + 𝑣)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑢﷮𝑑𝑥﷯ + 𝑑𝑣﷮𝑑𝑥﷯ Calculating 𝒅𝒖﷮𝒅𝒙﷯ 𝑢 = sin﷮𝑥﷯﷯﷮𝑥﷯ Taking log both sides log﷮𝑢﷯= log﷮ sin﷮𝑥﷯﷯﷮𝑥﷯﷯ log﷮𝑢﷯=𝑥 . log﷮ sin﷮𝑥 ﷯﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑( log﷮𝑢﷯)﷮𝑑𝑥﷯ = 𝑑 𝑥. log﷮ sin﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑢﷯﷯﷮𝑑𝑥﷯ . 𝑑𝑢﷮𝑑𝑢﷯ = 𝑑 𝑥 . log﷮ sin﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑢﷯﷯﷮𝑑𝑢﷯ . 𝑑𝑢﷮𝑑𝑥﷯ = 𝑑 𝑥 . log﷮ sin﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ 1﷮𝑢﷯ . 𝑑𝑢﷮𝑑𝑥﷯ = 𝑑 𝑥 . log﷮ sin﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ 1﷮𝑢﷯ . 𝑑𝑢﷮𝑑𝑥﷯ = 𝑑𝑥﷮𝑑𝑥﷯ . log⁡sin⁡𝑥 + 𝑑 log﷮ sin﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ × 𝑥 1﷮𝑢﷯ . 𝑑𝑢﷮𝑑𝑥﷯ = 1 . log⁡sin⁡𝑥 + 1﷮ sin﷮𝑥﷯﷯ . 𝑑 sin﷮𝑥﷯﷯﷮𝑑𝑥﷯﷯ 𝑥 1﷮𝑢﷯ . 𝑑𝑢﷮𝑑𝑥﷯ = log⁡sin⁡𝑥 + 1﷮ sin﷮𝑥﷯﷯ . cos﷮𝑥﷯﷯𝑥 1﷮𝑢﷯ . 𝑑𝑢﷮𝑑𝑥﷯ = log⁡sin⁡𝑥 + 𝑥 cos﷮𝑥﷯﷮ sin﷮𝑥﷯﷯﷯ 𝑑𝑢﷮𝑑𝑥﷯ = u log﷮ sin﷮𝑥+ cot﷮𝑥﷯﷯﷯﷯ 𝑑𝑢﷮𝑑𝑥﷯ = sin 𝑥﷯﷮𝑥﷯ log﷮ sin﷮𝑥+ cot﷮𝑥﷯﷯﷯﷯ Calculating 𝒅𝒗﷮𝒅𝒙﷯ 𝑣= sin﷮−1﷯﷮ ﷮𝑥﷯﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑣﷮𝑑𝑥﷯ = 𝑑 sin﷮−1﷯﷮ ﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ 𝑑𝑣﷮𝑑𝑥﷯ = 1﷮ ﷮1 − 𝑥﷯﷮2﷯﷯﷯ . 𝑑 ﷮𝑥﷯﷮𝑑𝑥﷯ 𝑑𝑣﷮𝑑𝑥﷯ = 1﷮ ﷮1 − 𝑥﷯﷯ . 1﷮2 ﷮𝑥﷯﷯ 𝑑𝑣﷮𝑑𝑥﷯ = 1﷮2 ﷮𝑥﷯ ﷮1 − 𝑥﷯﷯ 𝑑𝑣﷮𝑑𝑥﷯ = 1﷮2 ﷮𝑥 1 −𝑥﷯﷯ ﷯ 𝑑𝑣﷮𝑑𝑥﷯ = 1﷮2 ﷮𝑥 − 𝑥﷮2﷯﷯ ﷯ Now , 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑢﷮𝑑𝑥﷯ + 𝑑𝑣﷮𝑑𝑥﷯ Putting value of 𝑑𝑢﷮𝑑𝑥﷯ & 𝑑𝑣﷮𝑑𝑥﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝐬𝐢𝐧﷮𝒙﷯﷯﷮𝒙﷯ 𝒙 𝐜𝐨𝐭﷮𝒙﷯+ 𝐥𝐨𝐠﷮ 𝐬𝐢𝐧﷮𝒙﷯﷯﷯ + 𝟏﷮𝟐 ﷮𝒙 − 𝒙﷮𝟐﷯﷯ ﷯

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