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



Last updated at Jan. 3, 2020 by Teachoo
Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12
Transcript
Misc 23 If ๐ฆ=๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ) , โ 1 โค ๐ฅ โค 1, show that (1โ๐ฅ^2 ) (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 โ ๐ฅ ๐๐ฆ/๐๐ฅ โ๐2 ๐ฆ 0 . ๐ฆ=๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ) Let ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ=๐ก ๐ฆ=๐^๐ก Differentiating ๐ค.๐.๐ก.๐ฅ. ๐๐ฆ/๐๐ฅ = ๐(๐^๐ก )/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐(๐^๐ก )/๐๐ฅ ร ๐๐ก/๐๐ก ๐๐ฆ/๐๐ฅ = ๐(๐^๐ก )/๐๐ก ร ๐๐ก/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐^๐ก ร ๐๐ก/๐๐ฅ Putting value of ๐ก=ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ ๐๐ฆ/๐๐ฅ = ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) ร ๐(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ)/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) ร ๐ ๐(ใ๐๐๐ ใ^(โ1) ๐ฅ)/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) ร ๐ ((โ1)/โ(1 โ ๐ฅ^2 )) ๐๐ฆ/๐๐ฅ = (โ๐ ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ))/โ(1 โ ๐ฅ^2 ) โ(1 โ ๐ฅ^2 ) ๐๐ฆ/๐๐ฅ = โ๐ ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) โ(1 โ ๐ฅ^2 ) ๐๐ฆ/๐๐ฅ = โ๐ ๐ฆ Differentiating again w.r.t x ๐(โ(1 โ ๐ฅ^2 ) ๐๐ฆ/๐๐ฅ)/๐๐ฅ = d(โ๐๐ฆ)/๐๐ฅ ๐(โ(1 โ ๐ฅ^2 ))/๐๐ฅ ๐๐ฆ/๐๐ฅ+โ(1 โ ๐ฅ^2 ) ๐( ๐๐ฆ/๐๐ฅ)/๐๐ฅ = โ๐ ๐๐ฆ/๐๐ฅ (โ1)/(2โ(1 โ ๐ฅ^2 ))ร (1โ๐ฅ^2 )^โฒ ๐๐ฆ/๐๐ฅ+โ(1 โ ๐ฅ^2 ) (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = โ๐ ๐๐ฆ/๐๐ฅ (โ1)/(2โ(1 โ ๐ฅ^2 ))ร (โ2๐ฅ) ๐๐ฆ/๐๐ฅ+โ(1 โ ๐ฅ^2 ) (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = โ๐ ๐๐ฆ/๐๐ฅ ๐ฅ/โ(1 โ ๐ฅ^2 ) ๐๐ฆ/๐๐ฅ+โ(1 โ ๐ฅ^2 ) (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = โ๐ ๐๐ฆ/๐๐ฅ Multiplying โ(1 โ ๐ฅ^2 ) both sides ๐ฅ ๐๐ฆ/๐๐ฅ+(โ(1 โ ๐ฅ^2 ))^2 (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = โโ(1 โ ๐ฅ^2 ) ๐ ๐๐ฆ/๐๐ฅ ๐ฅ ๐๐ฆ/๐๐ฅ+(1โ๐ฅ^2 ) (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = โโ(1 โ ๐ฅ^2 ) ๐ ๐๐ฆ/๐๐ฅ ๐ฅ ๐๐ฆ/๐๐ฅ+(1โ๐ฅ^2 ) (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = โโ(1 โ ๐ฅ^2 ) ๐ ร(โ๐๐ฆ)/โ(1 โ ๐ฅ^2 ) ๐ฅ ๐๐ฆ/๐๐ฅ+(1โ๐ฅ^2 ) (๐^2 ๐ฆ)/(๐๐ฅ^2 ) = ๐^2 ๐ฆ Hence proved From (1) โ(1 โ ๐ฅ^2 ) ๐๐ฆ/๐๐ฅ = โ๐ ๐ฆ ๐๐ฆ/๐๐ฅ = (โ๐๐ฆ)/โ(1 โ ๐ฅ^2 )
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