Misc 10 - Differentiate xx + xa + ax + aa - Class 12 NCERT - Logarithmic Differentiation - Type 2

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Misc 10 Differentiate w.r.t. x the function, ๐‘ฅ๐‘ฅ + ๐‘ฅ๐‘Ž + ๐‘Ž๏ทฎ๐‘ฅ๏ทฏ+ ๐‘Ž๐‘Ž, for some fixed ๐‘Ž >0 and ๐‘ฅ> 0 Let ๐‘ฆ= ๐‘ฅ๐‘ฅ + ๐‘ฅ๐‘Ž + ๐‘Ž๏ทฎ๐‘ฅ๏ทฏ+ ๐‘Ž๐‘Ž Let u=๐‘ฅ๐‘ฅ , ๐‘ฃ=๐‘ฅ๐‘Ž , ๐‘ค= ๐‘Ž๏ทฎ๐‘ฅ๏ทฏ ๐‘ฆ=๐‘ข+๐‘ฃ+๐‘ค+๐‘Ž๐‘Ž Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘ข + ๐‘ฃ + ๐‘ค + ๐‘Ž๐‘Ž๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘ข๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ ๐‘ฃ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ+ ๐‘‘ ๐‘ค๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ ๐‘Ž๐‘Ž๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ+ ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 0 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ Calculating ๐’…๐’–๏ทฎ๐’…๐’™๏ทฏ ๐‘ข = ๐‘ฅ๏ทฎ๐‘ฅ๏ทฏ Taking log on both sides log๏ทฎ๐‘ข๏ทฏ= log๏ทฎ ๐‘ฅ๏ทฎ๐‘ฅ๏ทฏ๏ทฏ log๏ทฎ๐‘ข๏ทฏ=๐‘ฅ . log๏ทฎ๐‘ฅ๏ทฏ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘ log๏ทฎ๐‘ข๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘ฅ . log๏ทฎ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘ log๏ทฎ๐‘ข๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ข๏ทฏ = ๐‘‘ ๐‘ฅ . log๏ทฎ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘ log๏ทฎ๐‘ข๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ข๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . logโก๐‘ฅ + ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ log๏ทฎ๐‘ฅ๏ทฏ. ๐‘ฅ 1๏ทฎ๐‘ข๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ = logโก๐‘ฅ + 1๏ทฎ๐‘ฅ๏ทฏ . ๐‘ฅ 1๏ทฎ๐‘ข๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ = logโก๐‘ฅ + 1 ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ = u 1+ log ๐‘ฅ๏ทฏโก ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ฅ๏ทฎ๐‘ฅ๏ทฏ 1+ log ๐‘ฅ๏ทฏโก Calculating ๐’…๐’—๏ทฎ๐’…๐’™๏ทฏ ๐‘ฃ= ๐‘ฅ๏ทฎ๐‘Ž๏ทฏ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ= ๐‘‘ ๐‘ฅ๏ทฎ๐‘Ž๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ= ๐‘Ž ๐‘ฅ๏ทฎ๐‘Ž โˆ’1๏ทฏ Calculating ๐’…๐’˜๏ทฎ๐’…๐’™๏ทฏ ๐‘ค= ๐‘Ž๏ทฎ๐‘ฅ๏ทฏ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘Ž๏ทฎ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘Ž๏ทฎ๐‘ฅ๏ทฏ . log๏ทฎ๐‘Ž๏ทฏ Therefore ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐’™๏ทฎ๐’™๏ทฏ ๐Ÿ+ ๐’๐’๐’ˆ ๐’™๏ทฏ + ๐’‚ ๐’™๏ทฎ๐’‚ โˆ’๐Ÿ๏ทฏ + ๐’‚๏ทฎ๐’™๏ทฏ . ๐’๐’๐’ˆ๏ทฎ๐’‚๏ทฏ

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
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