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
Example 44 Differentiate w.r.t. x, the following function: (i) โ(3๐ฅ+2) + 1/โ(2๐ฅ^2+ 4) Let y = โ(3๐ฅ+2) + 1/โ(2๐ฅ^2+ 4 ) Differentiating ๐ค.๐.๐ก.๐ฅ ๐๐ฆ/๐๐ฅ = ๐(โ(3๐ฅ + 2) " + " 1/โ(2๐ฅ^2 + 4 ))/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐(โ(3๐ฅ + 2))/๐๐ฅ + ๐(1/โ(2๐ฅ^2 + 4 ))/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐(โ(3๐ฅ + 2))/๐๐ฅ + (๐(2๐ฅ^2 + 4)^((โ1)/2))/๐๐ฅ Calculating ๐(โ(3๐ฅ + 2))/๐๐ฅ & (๐(2๐ฅ^2 + 4)^((โ1)/2))/๐๐ฅ separately Calculating ๐(โ(๐๐ฑ + ๐))/๐ ๐ ๐(โ(3๐ฅ + 2))/๐๐ฅ = 1/(2โ(3๐ฅ + 2 )) ร ๐(3๐ฅ + 2)/๐๐ฅ = 1/(2โ(3๐ฅ + 2 )) ร (3+0) = 3/(2โ(3๐ฅ + 2 )) Calculating (๐ (๐๐^๐ + ๐)^((โ๐)/๐))/๐ ๐ (๐(2๐ฅ^2 + 4)^((โ1)/2))/๐๐ฅ = (โ1)/2 ใ(2๐ฅ^2+4)ใ^((โ1)/( 2) โ1) . ๐(2๐ฅ^2+ 4)/๐๐ฅ = (โ1)/2 (2๐ฅ^2+ 4)^((โ3)/( 2)) . (๐(2๐ฅ^2 )/๐๐ฅ + ๐(4)/๐๐ฅ) = (โ1)/2 (2๐ฅ^2+ 4)^((โ3)/( 2)) . (4๐ฅ+0) = (โ4๐ฅ)/2 (2๐ฅ^2+ 4)^((โ3)/( 2)) = (โ2๐ฅ)/(2๐ฅ^2+ 4)^(3/2) Hence, ๐๐ฆ/๐๐ฅ = ๐(โ(3๐ฅ+2))/๐๐ฅ + ๐(1/โ(2๐ฅ^2+ 4 ))/๐๐ฅ ๐ ๐/๐ ๐ = ๐/(๐โ(๐๐ + ๐ )) โ ๐๐/(๐๐^๐+ ๐)^(๐/( ๐)) Example 44 Differentiate w.r.t. x, the following function: (ii) ๐^(sec^2โก๐ฅ ) + 3cos^(โ1) ๐ฅ Let y = ๐^(sec^2โก๐ฅ ) + 3cos^(โ1) ๐ฅ Differentiating ๐ค.๐.๐ก.๐ฅ ๐๐ฆ/๐๐ฅ = ๐(๐^(sec^2โก๐ฅ )+ 3cos^(โ1) ๐ฅ" " )/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐(๐^(sec^2โก๐ฅ ) )/๐๐ฅ + ๐ (๐ใ๐๐๐ใ^(โ๐) ๐)/๐ ๐ ๐๐ฆ/๐๐ฅ = ๐^(sec^2โก๐ฅ ) ๐(sec^2โก๐ฅ )/๐๐ฅ + 3. ((โ๐)/โ(๐ โ๐^๐ )) ๐๐ฆ/๐๐ฅ = ๐^(sec^2โก๐ฅ ). 2 sec ๐ฅ . ๐ (ใ๐๐๐ ใโก๐ )/๐ ๐ โ 3/โ(1 โ๐ฅ^2 ) "As" ๐(๐^๐ฅ )/๐๐ฅ=๐^๐ฅ & ๐(ใ๐๐๐ ใ^(โ1)โก๐ฅ )/๐๐ฅ=(โ1)/โ(1 โใ ๐ฅใ^2 ) ๐๐ฆ/๐๐ฅ = ๐^(sec^2โก๐ฅ ). 2 sec ๐ฅ . ๐๐๐โก๐ .๐๐๐โก๐ โ 3/โ(1 โ ๐ฅ^2 ) ๐ ๐/๐ ๐ = ใ๐ ๐ใ^(ใ๐๐๐ใ^๐โก๐ ). ใ๐๐๐ใ^๐ ๐ .๐๐๐โก๐ โ ๐/โ(๐ โ ๐^๐ ) Example 44 Differentiate w.r.t. x, the following function: (iii) log7 (log x) y = log7 (log x) But we do not solve base 7. So, changing base y = log7 ( log x) y = ๐ฅ๐จ๐ โกใ(๐ฅ๐จ๐ โกใ๐)ใ ใ/๐ฅ๐จ๐ โก๐ Now, differentiating ๐๐ฆ/๐๐ฅ= ๐(((logโกใ(logโกใ7)ใ ใ)/(log 7))/๐๐ฅ ๐๐ฆ/๐๐ฅ= ๐/๐ฅ๐จ๐ โก๐ (๐(logโกใ(logโก๐ฅ ใ )) )/๐๐ฅ ๐๐ฆ/๐๐ฅ= 1/logโก7 . 1/logโก๐ฅ (๐(logโก๐ฅ))/๐๐ฅ ๐๐ฆ/๐๐ฅ= 1/logโก7 . 1/logโก๐ฅ . 1/๐ฅ ๐ ๐/๐ ๐= ๐/ใ๐ฑ ๐๐๐ใโกใ๐ ๐ฅ๐จ๐ โก๐ ใ (As log 7 is constant) ("As (log x)โฒ =" 1/๐ฅ) ("As (log x)โฒ =" 1/๐ฅ)
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