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
Examples
Example 2
Example 3
Example 4 Important
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Example 11
Example 12
Example 13
Example 14
Example 15
Example 16
Example 17 Important
Example 18
Example 19
Example 20 Important
Example 21
Example 22
Example 23
Example 24
Example 25
Example 26 Important
Example 27
Example 28
Example 29 Important
Example 30 Important
Example 31
Example 32 Important
Example 33 Important
Example 34
Example 35
Example 36 Important
Example 37 Important
Example 38
Example 39
Example 40
Example 41 You are here
Example 42 Important Not in Syllabus - CBSE Exams 2021
Example 43 Not in Syllabus - CBSE Exams 2021
Example 44 Important
Example 45 Important
Example 46
Example 47 Important
Example 48
About the Author
