Last updated at Jan. 3, 2020 by Teachoo
Check Full Chapter Explained - Continuity and Differentiability - https://you.tube/Chapter-5-Class-12-Continuity
Transcript
Example 41 If y = ใ๐ ๐๐ใ^(โ1) ๐ฅ, show that (1 โ ๐ฅ2) ๐2๐ฆ/๐๐ฅ2 โ ๐ฅ ๐๐ฆ/๐๐ฅ = 0 . We have ๐ฆ = ใ๐ ๐๐ใ^(โ1) ๐ฅ Differentiating ๐ค.๐.๐ก.๐ฅ ๐๐ฆ/๐๐ฅ = ๐(ใ๐ ๐๐ใ^(โ1) ๐ฅ)/๐๐ฅ ๐๐ฆ/๐๐ฅ = 1/โ(ใ1 โ ๐ฅใ^2 ) ๐ ๐/๐ ๐ = (ใ๐โ๐ใ^๐ )^((โ๐)/( ๐)) ("As " ๐(ใ๐ ๐๐ใ^(โ1) ๐ฅ)/๐๐ฅ " = " 1/โ(ใ1 โ ๐ฅใ^2 )) Again Differentiating ๐ค.๐.๐ก.๐ฅ ๐/๐๐ฅ (๐๐ฆ/๐๐ฅ) = (๐(ใ1 โ ๐ฅใ^2 )^((โ1)/( 2)))/๐๐ฅ (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 = (โ1)/( 2) (ใ1โ๐ฅใ^2 )^((โ1)/( 2) โ1) . ๐(ใ1 โ ๐ฅใ^2 )/๐๐ฅ (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 = (โ1)/( 2) (ใ1โ๐ฅใ^2 )^((โ3)/2 ). (0โ2๐ฅ) (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 = (โ1)/( 2) (ใ1โ๐ฅใ^2 )^((โ3)/2 ). (โ2๐ฅ) (๐ ^๐ ๐)/ใ๐ ๐ใ^๐ = ๐(ใ๐โ๐ใ^๐ )^((โ๐)/๐ ) Now, we need to prove (ใ1โ๐ฅใ^2 ) (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 โ ๐ฅ . ๐๐ฆ/๐๐ฅ = 0 Solving LHS (ใ1โ๐ฅใ^2 ) (๐^2 ๐ฆ)/ใ๐๐ฅใ^2 โ ๐ฅ . ๐๐ฆ/๐๐ฅ = (ใ1โ๐ฅใ^2 ) . (๐ฅใ (ใ1โ๐ฅใ^2 )ใ^((โ3)/2 ) ) โ ๐ฅ (ใ1โ๐ฅใ^2 )^((โ1)/( 2)) = ๐ฅใ (ใ1โ๐ฅใ^2 )ใ^(1+ ((โ3)/2) )โ๐ฅ (ใ1โ๐ฅใ^2 )^((โ1)/( 2)) = ๐ฅใ (ใ1โ๐ฅใ^2 )ใ^((โ1)/( 2))โ๐ฅ (ใ1โ๐ฅใ^2 )^((โ1)/( 2)) = 0 = RHS Hence proved
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