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

Transcript

Misc 23 If ๐ฆ=๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ) , โ 1 โค ๐ฅ โค 1, show that (1โ๐ฅ^2 ) (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 โ ๐ฅ ๐๐ฆ/๐๐ฅ โ๐2 ๐ฆ 0 . ๐ฆ=๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ) Let ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ=๐ก ๐ฆ=๐^๐ก Differentiating ๐ค.๐.๐ก.๐ฅ. ๐๐ฆ/๐๐ฅ = ๐(๐^๐ก )/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐(๐^๐ก )/๐๐ฅ ร ๐๐ก/๐๐ก ๐๐ฆ/๐๐ฅ = ๐(๐^๐ก )/๐๐ก ร ๐๐ก/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐^๐ก ร ๐๐ก/๐๐ฅ Putting value of ๐ก=ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ ๐๐ฆ/๐๐ฅ = ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) ร ๐(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ)/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) ร ๐ ๐(ใ๐๐๐ ใ^(โ1) ๐ฅ)/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ) ร ๐ ((โ1)/โ(1 โ ๐ฅ^2 )) ๐๐ฆ/๐๐ฅ = (โ๐ ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ))/โ(1 โ ๐ฅ^2 ) Again Differentiating ๐ค.๐.๐ก.๐ฅ. ๐/๐๐ฅ (๐๐ฆ/๐๐ฅ) = ๐/๐๐ฅ ((โ๐ ๐^(ใ๐ ๐๐๐ ใ^(โ1) ๐ฅ" " ))/โ(1 โ ๐ฅ^2 ))

Serial order wise