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

Ex 5.7, 5 Find the second order derivatives of the function ๐‘ฅ^3 logโก๐‘ฅ Let y = ๐‘ฅ^3 logโก๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘(๐‘ฅ^3 " " logโก๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘ฅ^3 )/๐‘‘๐‘ฅ .logโก๐‘ฅ + (๐‘‘(logโก๐‘ฅ))/๐‘‘๐‘ฅ .๐‘ฅ^3 using product rule in ๐‘ฅ^3 ๐‘™๐‘œ๐‘”โก๐‘ฅ . As (๐‘ข๐‘ฃ)โ€™= ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข where u = x3 & v = log x ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 3๐‘ฅ2 . logโก๐‘ฅ + 1/๐‘ฅ . ๐‘ฅ^3 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 3๐‘ฅ2 . logโก๐‘ฅ + ๐‘ฅ^2 Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = (๐‘‘ (3๐‘ฅ2 . logโก๐‘ฅ "+ " ๐‘ฅ^2))/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = (๐‘‘ (3๐‘ฅ2 . logโก๐‘ฅ))/๐‘‘๐‘ฅ + (๐‘‘ (๐‘ฅ^2))/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = (3 ๐‘‘(๐‘ฅ2 . logโก๐‘ฅ))/๐‘‘๐‘ฅ + 2 ๐‘ฅ using product rule in ๐‘ฅ^2 ๐‘™๐‘œ๐‘”โก๐‘ฅ . As (๐‘ข๐‘ฃ)โ€™= ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = 3 . ((๐‘‘(๐‘ฅ)^2)/๐‘‘๐‘ฅ .logโกใ€–๐‘ฅ+(๐‘‘(logโกใ€–๐‘ฅ)ใ€—)/๐‘‘๐‘ฅใ€—. ๐‘ฅ^2 ) + 2 ๐‘ฅ = 3(2๐‘ฅ . logโกใ€–๐‘ฅ+ 1/๐‘ฅใ€—. ๐‘ฅ^2 ) + 2 ๐‘ฅ = 3 (2๐‘ฅ . logโก๐‘ฅ+ ๐‘ฅ) + 2 ๐‘ฅ = 6 ๐‘ฅ logโก๐‘ฅ + 3๐‘ฅ + 2๐‘ฅ = 6 ๐‘ฅ log โก๐‘ฅ+5๐‘ฅ = x (6 logโกใ€–๐‘ฅ+5ใ€— ) = ๐‘ฅ (5+6 logโก๐‘ฅ ) Hence , (๐’…^๐Ÿ ๐’š)/(๐’…๐’™^๐Ÿ ) = ๐’™ (๐Ÿ“+๐Ÿ” ๐’๐’๐’ˆโก๐’™ )

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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.