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Transcript

Misc 10 Differentiate w.r.t. x the function, ๐‘ฅ๐‘ฅ + ๐‘ฅ๐‘Ž + ๐‘Ž^๐‘ฅ+ ๐‘Ž๐‘Ž, for some fixed ๐‘Ž >0 and ๐‘ฅ> 0Let ๐‘ฆ= ๐‘ฅ๐‘ฅ + ๐‘ฅ๐‘Ž + ๐‘Ž^๐‘ฅ+ ๐‘Ž๐‘Ž And let u=๐‘ฅ๐‘ฅ , ๐‘ฃ=๐‘ฅ๐‘Ž , ๐‘ค=๐‘Ž^๐‘ฅ Now, ๐’š=๐’–+๐’—+๐’˜+๐’‚๐’‚ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘ข + ๐‘ฃ + ๐‘ค + ๐‘Ž๐‘Ž)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘ข)/๐‘‘๐‘ฅ +๐‘‘(๐‘ฃ)/๐‘‘๐‘ฅ+๐‘‘(๐‘ค)/๐‘‘๐‘ฅ + ๐‘‘(๐‘Ž๐‘Ž)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ +๐‘‘๐‘ฃ/๐‘‘๐‘ฅ+๐‘‘๐‘ค/๐‘‘๐‘ฅ + 0 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ + ๐‘‘๐‘ค/๐‘‘๐‘ฅ Calculating ๐’…๐’–/๐’…๐’™ ๐‘ข =๐‘ฅ^๐‘ฅ Taking log on both sides logโก๐‘ข=logโกใ€–๐‘ฅ^๐‘ฅ ใ€— logโก๐‘ข=๐‘ฅ .logโก๐‘ฅ (๐‘Ž^๐‘Ž ๐‘–๐‘  ๐‘๐‘œ๐‘›๐‘ ๐‘ก๐‘Ž๐‘›๐‘ก) Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ฅ = ๐‘‘(๐‘ฅ .ใ€– logใ€—โก๐‘ฅ )/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ฅ . ๐‘‘๐‘ข/๐‘‘๐‘ข = ๐‘‘(๐‘ฅ .ใ€– logใ€—โก๐‘ฅ )/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = ๐‘‘๐‘ฅ/๐‘‘๐‘ฅ . logโก๐‘ฅ + (๐‘‘(ใ€– logใ€—โก๐‘ฅ))/๐‘‘๐‘ฅ. ๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = logโก๐‘ฅ + 1/๐‘ฅ . ๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = logโก๐‘ฅ + 1 ๐‘‘๐‘ข/๐‘‘๐‘ฅ = u (1+ log ๐‘ฅ)โก ๐’…๐’–/๐’…๐’™ = ๐’™^๐’™ (๐Ÿ+ ๐’๐’๐’ˆ ๐’™)โก Calculating ๐’…๐’—/๐’…๐’™ ๐‘ฃ=๐‘ฅ^๐‘Ž Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ= ๐‘‘(๐‘ฅ^๐‘Ž )/๐‘‘๐‘ฅ ๐’…๐’—/๐’…๐’™= ๐’‚๐’™^(๐’‚ โˆ’๐Ÿ) Calculating ๐’…๐’˜/๐’…๐’™ ๐‘ค=๐‘Ž^๐‘ฅ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ค/๐‘‘๐‘ฅ = ๐‘‘(๐‘Ž^๐‘ฅ )/๐‘‘๐‘ฅ ๐‘‘๐‘ค/๐‘‘๐‘ฅ = ๐‘Ž^๐‘ฅ .logโก๐‘Ž Therefore, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ + ๐‘‘๐‘ค/๐‘‘๐‘ฅ = ๐’™^๐’™ (๐Ÿ+ ๐’๐’๐’ˆ ๐’™) + ๐’‚๐’™^(๐’‚ โˆ’๐Ÿ) + ๐’‚^๐’™ .๐’๐’๐’ˆโก๐’‚

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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, Social Science, Physics, Chemistry, Computer Science at Teachoo.