Misc 11 - Differentiate x x2-3 + (x - 3)x2 - Chapter 5 Class 12 - Logarithmic Differentiation - Type 2

Slide10.JPG
Slide11.JPG Slide12.JPG Slide13.JPG Slide14.JPG

  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
Ask Download

Transcript

Misc 11 (Method 1) Differentiate w.r.t. x the function, 𝑥﷮ 𝑥﷮2﷯− 3﷯+(𝑥−3) 𝑥﷮2﷯, for 𝑥 > 3 Calculating 𝒅𝒖﷮𝒅𝒙﷯ 𝑢 = 𝑥﷮ 𝑥﷮2﷯− 3﷯ Taking log on both sides log 𝑢= log﷮ 𝑥﷮ 𝑥﷮2﷯− 3﷯﷯ log 𝑢= (𝑥﷮2﷯− 3). log﷮𝑥﷯ Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑 log﷮𝑢﷯﷯﷮𝑑𝑥﷯ = 𝑑 (𝑥﷮2﷯− 3) log﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑢﷯﷯﷮𝑑𝑥﷯ . 𝑑𝑢﷮𝑑𝑢﷯ = 𝑑 (𝑥﷮2﷯ − 3) log﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑𝑢﷮𝑑𝑥﷯ = 𝑥﷮ 𝑥﷮2﷯− 3﷯ 2𝑥 . log﷮𝑥﷯+ 𝑥﷮2﷯− 3﷮𝑥﷯﷯ Calculating 𝒅𝒗﷮𝒅𝒙﷯ 𝑣= (𝑥−3) 𝑥﷮2﷯ Taking log on both sides log 𝑣= log﷮(𝑥−3) 𝑥﷮2﷯﷯ log 𝑣= 𝑥﷮2﷯ . log﷮ (𝑥−3)﷯ Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑 log﷮𝑣﷯﷯﷮𝑑𝑥﷯ = 𝑑 𝑥﷮2﷯. log﷮ (𝑥−3)﷯﷯ ﷮𝑑𝑥﷯ 𝑑 log﷮𝑣﷯﷯﷮𝑑𝑥﷯ . 𝑑𝑣﷮𝑑𝑣﷯ = 𝑑 𝑥﷮2﷯. log﷮ (𝑥−3)﷯﷯ ﷮𝑑𝑥﷯ 𝑑 log﷮𝑣﷯﷯﷮𝑑𝑣﷯ . 𝑑𝑣﷮𝑑𝑥﷯ = 𝑑 𝑥﷮2﷯. log﷮ (𝑥−3)﷯﷯ ﷮𝑑𝑥﷯ 1﷮𝑣﷯ . 𝑑𝑣﷮𝑑𝑥﷯ = 𝑑 𝑥﷮2﷯. log﷮ (𝑥−3)﷯﷯ ﷮𝑑𝑥﷯ 1﷮𝑣﷯ . 𝑑𝑣﷮𝑑𝑥﷯ = 𝑑 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ . log (𝑥−3) + 𝑑 log (𝑥 − 3)﷯﷮𝑑𝑥﷯ . 𝑥﷮2﷯ 1﷮𝑣﷯ . 𝑑𝑣﷮𝑑𝑥﷯ = 2𝑥 . log (𝑥−3) + 1﷮ 𝑥 − 3﷯﷯. 𝑑(𝑥 − 3) ﷮𝑑𝑥﷯ . 𝑥﷮2﷯ 1﷮𝑣﷯ . 𝑑𝑣﷮𝑑𝑥﷯ = 2𝑥 . log (𝑥−3) + 1﷮ 𝑥 − 3﷯﷯ . 𝑥﷮2﷯ 1﷮𝑣﷯ . 𝑑𝑣﷮𝑑𝑥﷯ = 2𝑥. log (𝑥−3) + 𝑥﷮2﷯﷮𝑥 −3﷯ 𝑑𝑣﷮𝑑𝑥﷯ = 𝑣 2𝑥. log (𝑥−3) + 𝑥﷮2﷯﷮𝑥 −3﷯﷯ 𝑑𝑣﷮𝑑𝑥﷯ = (𝑥−3) 𝑥﷮2﷯ 2𝑥. log (𝑥−3) + 𝑥﷮2﷯﷮𝑥 −3﷯﷯ Now, 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑢﷮𝑑𝑥﷯ + 𝑑𝑣﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥﷮ 𝑥﷮2﷯− 3﷯ 2𝑥 . log﷮𝑥﷯+ 𝑥﷮2﷯− 3﷮𝑥﷯﷯ + (𝑥−3) 𝑥﷮2﷯ 2𝑥. log (𝑥−3) + 𝑥﷮2﷯﷮𝑥 −3﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒙﷮ 𝒙﷮𝟐﷯− 𝟑﷯ 𝒙﷮𝟐﷯− 𝟑﷮𝒙﷯+𝟐𝒙 𝐥𝐨𝐠﷮𝒙﷯﷯ + (𝒙−𝟑) 𝒙﷮𝟐﷯ 𝒙﷮𝟐﷯﷮𝒙 −𝟑﷯+𝟐𝒙 . 𝐥𝐨𝐠﷮ 𝒙 −𝟑﷯﷯﷯

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
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.
Jail