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Ex 5.5, 2 - Differentiate root (x-1)(x-2)/(x-3)(x-4)(x-5) - Logarithmic Differentiation - Type 1

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.5, 2 Differentiate the functions in, ﷮ 𝑥 − 1﷯(𝑥 − 2)﷮ 𝑥 − 3﷯ 𝑥 − 4﷯(𝑥 − 5)﷯﷯ Let 𝑦= ﷮ 𝑥 − 1﷯(𝑥 − 2)﷮ 𝑥 − 3﷯ 𝑥 − 4﷯(𝑥 − 5)﷯﷯ 𝑦= 𝑥 − 1﷯(𝑥 − 2)﷮ 𝑥 − 3﷯ 𝑥 − 4﷯(𝑥 − 5)﷯﷯﷮ 1﷮2﷯﷯ Taking log both sides log⁡𝑦 = log 𝑥 − 1﷯(𝑥 − 2)﷮ 𝑥 − 3﷯ 𝑥 − 4﷯(𝑥 − 5)﷯﷯﷮ 1﷮2﷯﷯ log⁡𝑦 = 1﷮2﷯ log 𝑥 − 1﷯(𝑥 − 2)﷮ 𝑥 − 3﷯ 𝑥 − 4﷯(𝑥 − 5)﷯﷯ log⁡𝑦 = 1﷮2﷯ log﷮ 𝑥−1﷯(𝑥−2)﷯− log﷮ 𝑥−3﷯ 𝑥−4﷯(𝑥−5)﷯﷯﷯﷯ log⁡𝑦 = 1﷮2﷯. log 𝑥+1﷯ + log (𝑥−2)﷯ − log﷮ 𝑥−3﷯﷯+ log﷮ 𝑥−4﷯+ log﷮ 𝑥−5﷯﷯﷯﷯ log⁡𝑦 = 1﷮2﷯ . log 𝑥+1﷯ + log (𝑥+2) − log﷮ 𝑥−3﷯﷯+ log﷮ 𝑥−4﷯+ log﷮ 𝑥−5﷯﷯﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑 log⁡𝑦﷯﷮𝑑𝑥﷯ = 𝑑 1﷮2﷯ . log﷮ 𝑥+1﷯ + log﷮ (𝑥+2) − log﷮ 𝑥−3﷯﷯+ log﷮ 𝑥−4﷯+ log﷮ 𝑥−5﷯﷯﷯﷯﷯﷯﷮𝑑𝑥﷯ 𝑑 log⁡𝑦﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑦﷯﷯ = 1﷮2﷯ 𝑑 log ﷮ 𝑥 + 1﷯ + log﷮(𝑥 + 2) − log﷮ 𝑥 − 3﷯﷯ + log﷮ 𝑥 − 4﷯ + log﷮ 𝑥 − 5﷯﷯﷯﷯﷯﷯﷮𝑑𝑥﷯﷯ 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮2﷯ 1﷮𝑥 + 1﷯+ 1﷮𝑥 + 2﷯− 1﷮𝑥 − 3﷯− 1﷮𝑥 − 4﷯− 1﷮𝑥 − 5﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮2﷯ 𝑦 1﷮𝑥 − 1﷯+ 1﷮𝑥 − 2﷯− 1﷮𝑥 − 3﷯− 1﷮𝑥 − 4﷯− 1﷮𝑥 − 5﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝟏﷮𝟐﷯ ﷮ 𝒙−𝟏﷯(𝒙−𝟐)﷮ 𝒙−𝟑﷯ 𝒙−𝟒﷯(𝒙−𝟓)﷯﷯ 𝟏﷮𝒙 − 𝟏﷯+ 𝟏﷮𝒙 − 𝟐﷯− 𝟏﷮𝒙 − 𝟑﷯− 𝟏﷮𝒙 − 𝟒﷯− 𝟏﷮𝒙 − 𝟓﷯﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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