Ex 5.5, 12 - Find dy/dx, xy + yx = 1 - Class 12 CBSE NCERT

Ex 5.5, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.5, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 3
Ex 5.5, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 4
Ex 5.5, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 5 Ex 5.5, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 6 Ex 5.5, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 7

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Ex 5.5, 12 Find ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ of the functions in, ๐‘ฅ^๐‘ฆ + ๐‘ฆ^๐‘ฅ = 1 ๐‘ฅ^๐‘ฆ + ๐‘ฆ^๐‘ฅ = 1 Let ๐‘ข = ๐‘ฅ^๐‘ฆ , ๐‘ฃ = ๐‘ฆ^๐‘ฅ Hence, ๐‘ข+๐‘ฃ=1 Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. (๐‘‘(๐‘ฃโกใ€–+ ๐‘ขใ€—))/๐‘‘๐‘ฅ = ๐‘‘(1)/๐‘‘๐‘ฅ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ + ๐‘‘๐‘ข/๐‘‘๐‘ฅ = 0 (Derivative of constant is 0) Calculating ๐’…๐’—/๐’…๐’™ ๐‘ฃ=๐‘ฅ^๐‘ฆ Taking log both sides logโก๐‘ฃ=logโกใ€– (๐‘ฅ^๐‘ฆ)" " ใ€— logโก๐‘ฃ=ใ€–๐‘ฆ. logใ€—โก๐‘ฅ" " Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ (๐‘‘(logโก๐‘ฃ))/๐‘‘๐‘ฅ = (๐‘‘(๐‘ฆ . logโก๐‘ฅ))/๐‘‘๐‘ฅ (๐‘‘(logโก๐‘ฃ))/๐‘‘๐‘ฅ (๐‘‘๐‘ฃ/๐‘‘๐‘ฃ) = ๐‘‘(๐‘ฆ logโก๐‘ฅ )/๐‘‘๐‘ฅ (๐‘‘(logโก๐‘ฃ))/๐‘‘๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ๐‘‘(ใ€–๐‘ฆ logใ€—โก๐‘ฅ )/๐‘‘๐‘ฅ 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ( ๐‘‘(ใ€–๐‘ฆ logใ€—โก๐‘ฅ ))/๐‘‘๐‘ฅ 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ( ๐‘‘(๐‘ฆ))/๐‘‘๐‘ฅ . logโก๐‘ฅ + (๐‘‘ (logโก๐‘ฅ))/๐‘‘๐‘ฅ . ๐‘ฆ 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ( ๐‘‘๐‘ฆ)/๐‘‘๐‘ฅ . logโก๐‘ฅ + 1/๐‘ฅ . ๐‘ฆ 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ( ๐‘‘๐‘ฆ)/๐‘‘๐‘ฅ logโก๐‘ฅ + ๐‘ฆ/๐‘ฅ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = v (log ๐‘ฅ+๐‘ฆ/๐‘ฅ) Putting value of ๐‘ฃ = ๐‘ฅ^๐‘ฆ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฅ^๐‘ฆ (logโกใ€–๐‘ฅ+ ๐‘ฆ/๐‘ฅใ€— ) ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฅ^๐‘ฆ.log ๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + ๐‘ฅ^๐‘ฆ. ๐‘ฆ/๐‘ฅ Using product Rule As (๐‘ข๐‘ฃ)โ€™ = ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข Calculating ๐’…๐’–/๐’…๐’™ ๐‘ข = ๐‘ฆ^๐‘ฅ Taking log both sides logโก๐‘ข=logโกใ€– (๐‘ฆ^๐‘ฅ)" " ใ€— logโก๐‘ข=ใ€–๐‘ฅ . logใ€—โก๐‘ฆ" " Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ (๐‘‘(logโก๐‘ข))/๐‘‘๐‘ฅ = (๐‘‘(๐‘ฅ . logโก๐‘ฆ))/๐‘‘๐‘ฅ (๐‘‘(logโก๐‘ข))/๐‘‘๐‘ฅ (๐‘‘๐‘ข/๐‘‘๐‘ข) = ๐‘‘(๐‘ฅ.logโก๐‘ฆ )/๐‘‘๐‘ฅ (๐ด๐‘  logโกใ€–(๐‘Ž^๐‘)ใ€—=๐‘ logโก๐‘Ž ) (๐‘‘(logโก๐‘ข))/๐‘‘๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ) = (๐‘‘ (๐‘ฅ . logโก๐‘ฆ))/๐‘‘๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = (๐‘‘ (๐‘ฅ . logโก๐‘ฆ ))/๐‘‘๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = (๐‘‘ (๐‘ฅ))/๐‘‘๐‘ฅ . logโก๐‘ฆ + (๐‘‘(logโก๐‘ฆ))/๐‘‘๐‘ฅ . ๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = 1 . logโก๐‘ฆ + ๐‘ฅ. ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฆ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = logโก๐‘ฆ + ๐‘ฅ. ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = logโก๐‘ฆ + ๐‘ฅ. 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ Using product Rule As (๐‘ข๐‘ฃ)โ€™ = ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข ๐‘‘๐‘ข/๐‘‘๐‘ฅ = ๐‘ข(logโก๐‘ฆ " + " ๐‘ฅ/๐‘ฆ " . " (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ) ๐‘‘๐‘ข/๐‘‘๐‘ฅ " = " ๐‘ฆ^๐‘ฅ (logโก๐‘ฆ " + " ๐‘ฅ/๐‘ฆ " . " (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ) ๐‘‘๐‘ข/๐‘‘๐‘ฅ = ๐‘ฆ^๐‘ฅ logโก๐‘ฆ+๐‘ฆ^(๐‘ฅ โˆ’1). ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ Now, ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ + ๐‘‘๐‘ข/๐‘‘๐‘ฅ = 0 Putting value of ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ & ๐‘‘๐‘ข/๐‘‘๐‘ฅ (๐‘ฅ^๐‘ฆ logโกใ€–๐‘ฅ ใ€— ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ +๐‘ฅ^๐‘ฆ ๐‘ฆ/๐‘ฅ) + (๐‘ฆ^๐‘ฅ logโกใ€–๐‘ฆ+๐‘ฆ^(๐‘ฅ โˆ’1) ใ€—.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = 0 ๐‘ฅ^๐‘ฆ logโกใ€–๐‘ฅ ใ€— ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ +๐‘ฅ^(๐‘ฆโˆ’1) ๐‘ฆ + ๐‘ฆ^๐‘ฅ logโกใ€–๐‘ฆ+๐‘ฆ^(๐‘ฅ โˆ’1) ใ€—.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 ๐‘ฅ^๐‘ฆ logโกใ€–๐‘ฅ ใ€— ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฆ^(๐‘ฅ โˆ’1).๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=โˆ’(๐‘ฅ^(๐‘ฆโˆ’1) ๐‘ฆ+๐‘ฆ^๐‘ฅ logโก๐‘ฆ ) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ(๐‘ฅ^๐‘ฆ logโกใ€–๐‘ฅ ใ€—+๐‘ฆ^(๐‘ฅ โˆ’1).๐‘ฅ)=โˆ’(๐‘ฅ^(๐‘ฆโˆ’1) ๐‘ฆ+๐‘ฆ^๐‘ฅ logโก๐‘ฆ ) ๐’…๐’š/๐’…๐’™ = (โˆ’(๐’™^(๐’šโˆ’๐Ÿ) ๐’š + ๐’š^๐’™ ๐’๐’๐’ˆโก๐’š ))/((๐’™^๐’š ๐’๐’๐’ˆโกใ€–๐’™ ใ€—+ ๐’š^(๐’™ โˆ’๐Ÿ).๐’™))

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo