Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

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

Transcript

Example 47 For a positive constant a find ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ , where ๐‘ฆ = ๐‘Ž^(๐‘ก+1/๐‘ก) , and ๐‘ฅ =(๐‘ก+1/๐‘ก)^2 Here ๐’…๐’š/๐’…๐’™ = (๐’…๐’š/๐’…๐’•)/(๐’…๐’™/๐’…๐’•) Calculating ๐’…๐’š/๐’…๐’• ๐‘ฆ=๐‘Ž^(๐‘ก + 1/๐‘ก) Differentiating ๐‘ค.๐‘Ÿ.๐‘ก. t ๐’…๐’š/๐’…๐’• = ๐’…(๐’‚^((๐’• + ๐Ÿ/๐’•) ) )/๐’…๐’• ๐‘‘๐‘ฆ/๐‘‘๐‘ก = ๐‘Ž^((๐‘ก + 1/๐‘ก) ) .logโก๐‘Ž.๐‘‘(๐‘ก + 1/๐‘ก)/๐‘‘๐‘ก ๐‘‘๐‘ฆ/๐‘‘๐‘ก = ๐‘Ž^((๐‘ก + 1/๐‘ก) ) .logโก๐‘Ž.(1+(โˆ’1) ๐‘ก^(โˆ’2) ) ๐’…๐’š/๐’…๐’• = ๐’‚^((๐’• + ๐Ÿ/๐’•) ) .๐’๐’๐’ˆโก๐’‚.(๐Ÿโˆ’๐Ÿ/๐’•^๐Ÿ ) "As " ๐‘‘(๐‘Ž^๐‘ฅ )/๐‘‘๐‘ฅ " = " ๐‘Ž^๐‘ฅ.๐‘™๐‘œ๐‘”โก๐‘Ž Calculating ๐’…๐’™/๐’…๐’• ๐‘ฅ=(๐‘ก+1/๐‘ก)^๐‘Ž Differentiating ๐‘ค.๐‘Ÿ.๐‘ก. t ๐‘‘๐‘ฅ/๐‘‘๐‘ก = ๐‘‘((๐‘ก + 1/๐‘ก)^(๐‘Ž ) )/๐‘‘๐‘ก ๐‘‘๐‘ฅ/๐‘‘๐‘ก = a (๐‘ก+1/๐‘ก)^(๐‘Ž โˆ’1 ) . ๐‘‘(๐‘ก + 1/๐‘ก)/๐‘‘๐‘ก ๐‘‘๐‘ฅ/๐‘‘๐‘ก = a (๐‘ก+1/๐‘ก)^(๐‘Ž โˆ’1 ) . (๐‘‘(๐‘ก)/๐‘‘๐‘ก + ๐‘‘(1/๐‘ก)/๐‘‘๐‘ก) ๐‘‘๐‘ฅ/๐‘‘๐‘ก = a (๐‘ก+1/๐‘ก)^(๐‘Ž โˆ’1 ) . (1+ ๐‘‘(๐‘ก^(โˆ’1) )/๐‘‘๐‘ก) ๐‘‘๐‘ฅ/๐‘‘๐‘ก = a ๐‘^(๐‘Ž โˆ’1 ) . ๐‘‘(๐‘)/๐‘‘๐‘ก ๐‘‘๐‘ฅ/๐‘‘๐‘ก = a (๐‘ก+1/๐‘ก)^(๐‘Ž โˆ’1 ) . (1+(โˆ’1) ใ€– ๐‘กใ€—^(โˆ’2) ) ๐‘‘๐‘ฅ/๐‘‘๐‘ก = a (๐‘ก+1/๐‘ก)^(๐‘Ž โˆ’1 ) . (1โˆ’ 1/๐‘ก^2 ) Calculating ๐’…๐’š/๐’…๐’™ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘๐‘ฆ/๐‘‘๐‘ก)/(๐‘‘๐‘ฅ/๐‘‘๐‘ก) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘Ž^(๐‘ก + 1/๐‘ก) . logโกใ€–๐‘Ž ใ€— ร— (1 โˆ’ 1/๐‘ก^2 ))/(๐‘Ž(๐‘ก + 1/๐‘ก)^(๐‘Ž โˆ’ 1) (1 โˆ’ 1/๐‘ก^2 ).) ๐’…๐’š/๐’…๐’™ = (๐’‚^(๐’• + ๐Ÿ/๐’•) . ๐’๐’๐’ˆโกใ€–๐’‚ ใ€—)/(๐’‚(๐’• + ๐Ÿ/๐’•)^(๐’‚ โˆ’ ๐Ÿ) )

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.