Ex 5.5, 15 - Find dy/dx of xy = e(x - y) - Class 12 - 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, 15 Find ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ of the functions in, ๐‘ฅ๐‘ฆ= ๐‘’^((๐‘ฅ โˆ’๐‘ฆ)) Given ๐‘ฅ๐‘ฆ= ๐‘’^((๐‘ฅ โˆ’๐‘ฆ)) Taking log both sides log (๐‘ฅ๐‘ฆ) = log ๐‘’^((๐‘ฅ โˆ’๐‘ฆ))(As ๐‘™๐‘œ๐‘”โก(๐‘Ž^๐‘ )=๐‘ . ๐‘™๐‘œ๐‘”โก๐‘Ž) log (๐‘ฅ๐‘ฆ) = (๐‘ฅ โˆ’๐‘ฆ) log ๐‘’ ("As " ๐‘™๐‘œ๐‘”โก๐‘’=1) log ๐‘ฅ+logโก๐‘ฆ = (๐‘ฅ โˆ’๐‘ฆ) (1) log ๐‘ฅ+logโก๐‘ฆ = (๐‘ฅ โˆ’๐‘ฆ) Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(log ๐‘ฅ + logโก๐‘ฆ )/๐‘‘๐‘ฅ = (๐‘‘(๐‘ฅ โˆ’ ๐‘ฆ))/๐‘‘๐‘ฅ ๐‘‘(log ๐‘ฅ)/๐‘‘๐‘ฅ + ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ = ๐‘‘(๐‘ฅ)/๐‘‘๐‘ฅ โˆ’ ๐‘‘(๐‘ฆ)/๐‘‘๐‘ฅ 1/๐‘ฆ + ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฆ = 1 โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ 1/๐‘ฆ + ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1 โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ 1/๐‘ฆ + 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1 โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1 โˆ’ 1/๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ (1/๐‘ฆ +1) = ("1 โˆ’ " 1/๐‘ฅ) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ((1 + ๐‘ฆ)/๐‘ฆ) = ((๐‘ฅ โˆ’ 1)/๐‘ฅ) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ((๐‘ฅ โˆ’ 1)/๐‘ฆ) . (๐‘ฆ/(1 + ๐‘ฆ)) ๐’…๐’š/๐’…๐’™ = ๐’š(๐’™ โˆ’ ๐Ÿ)/๐’™(๐Ÿ + ๐’š)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.