Example 42 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 42 For a positive constant a find 𝑑𝑦/𝑑𝑥 , where 𝑦 = 𝑎^(𝑡+1/𝑡) , and 𝑥 =(𝑡+1/𝑡)^2 Here 𝒅𝒚/𝒅𝒙 = (𝒅𝒚/𝒅𝒕)/(𝒅𝒙/𝒅𝒕) Calculating 𝒅𝒚/𝒅𝒕 𝑦=𝑎^(𝑡 + 1/𝑡) Differentiating 𝑤.𝑟.𝑡. t 𝒅𝒚/𝒅𝒕 = 𝒅(𝒂^((𝒕 + 𝟏/𝒕) ) )/𝒅𝒕 𝑑𝑦/𝑑𝑡 = 𝑎^((𝑡 + 1/𝑡) ) .log𝑎.𝑑(𝑡 + 1/𝑡)/𝑑𝑡 𝑑𝑦/𝑑𝑡 = 𝑎^((𝑡 + 1/𝑡) ) .log𝑎.(1+(−1) 𝑡^(−2) ) 𝒅𝒚/𝒅𝒕 = 𝒂^((𝒕 + 𝟏/𝒕) ) .𝒍𝒐𝒈𝒂.(𝟏−𝟏/𝒕^𝟐 ) "As " 𝑑(𝑎^𝑥 )/𝑑𝑥 " = " 𝑎^𝑥.𝑙𝑜𝑔𝑎 Calculating 𝒅𝒙/𝒅𝒕 𝑥=(𝑡+1/𝑡)^𝑎 Differentiating 𝑤.𝑟.𝑡. t 𝑑𝑥/𝑑𝑡 = 𝑑((𝑡 + 1/𝑡)^(𝑎 ) )/𝑑𝑡 𝑑𝑥/𝑑𝑡 = a (𝑡+1/𝑡)^(𝑎 −1 ) . 𝑑(𝑡 + 1/𝑡)/𝑑𝑡 𝑑𝑥/𝑑𝑡 = a (𝑡+1/𝑡)^(𝑎 −1 ) . (𝑑(𝑡)/𝑑𝑡 + 𝑑(1/𝑡)/𝑑𝑡) 𝑑𝑥/𝑑𝑡 = a (𝑡+1/𝑡)^(𝑎 −1 ) . (1+ 𝑑(𝑡^(−1) )/𝑑𝑡) 𝑑𝑥/𝑑𝑡 = a 𝑝^(𝑎 −1 ) . 𝑑(𝑝)/𝑑𝑡 𝑑𝑥/𝑑𝑡 = a (𝑡+1/𝑡)^(𝑎 −1 ) . (1+(−1) 〖 𝑡〗^(−2) ) 𝑑𝑥/𝑑𝑡 = a (𝑡+1/𝑡)^(𝑎 −1 ) . (1− 1/𝑡^2 ) Calculating 𝒅𝒚/𝒅𝒙 𝑑𝑦/𝑑𝑥 = (𝑑𝑦/𝑑𝑡)/(𝑑𝑥/𝑑𝑡) 𝑑𝑦/𝑑𝑥 = (𝑎^(𝑡 + 1/𝑡) . log〖𝑎 〗 × (1 − 1/𝑡^2 ))/(𝑎(𝑡 + 1/𝑡)^(𝑎 − 1) (1 − 1/𝑡^2 ).) 𝒅𝒚/𝒅𝒙 = (𝒂^(𝒕 + 𝟏/𝒕) . 𝒍𝒐𝒈〖𝒂 〗)/(𝒂(𝒕 + 𝟏/𝒕)^(𝒂 − 𝟏) )
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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