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

Transcript

Misc 14 If ๐‘ฅ โˆš(1+๐‘ฆ)+๐‘ฆ โˆš(1+๐‘ฅ) = 0 , for โ€“1 < ๐‘ฅ < 1, prove that ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’1)/(1 + ๐‘ฅ)2 ๐‘ฅ โˆš(1+๐‘ฆ)+๐‘ฆ โˆš(1+๐‘ฅ) = 0 ๐‘ฅ โˆš(1+๐‘ฆ) = โ€“ ๐‘ฆ โˆš(1+๐‘ฅ) Squaring both sides (๐‘ฅโˆš(1+๐‘ฆ) )^2 = (โˆ’๐‘ฆ โˆš(1+๐‘ฅ))^2 ๐‘ฅ^2 (โˆš(1+๐‘ฆ ) )^2 = (โˆ’๐‘ฆ)^2 (โˆš(1+๐‘ฅ))^2 ๐‘ฅ^2 (1+๐‘ฆ) = ๐‘ฆ^2 (1+๐‘ฅ) ๐‘ฅ^2+๐‘ฅ^2 ๐‘ฆ = ๐‘ฆ^2 + ๐‘ฆ^2 ๐‘ฅ ๐‘ฅ^2 โˆ’ ๐‘ฆ^2 = xy2 โˆ’ x2y (๐‘ฅ โˆ’๐‘ฆ) (๐‘ฅ+๐‘ฆ)=๐‘ฅ๐‘ฆ (๐‘ฆ โˆ’๐‘ฅ) โˆ’ (๐‘ฆ โˆ’๐‘ฅ) (๐‘ฅ+๐‘ฆ)=๐‘ฅ๐‘ฆ (๐‘ฆ โˆ’๐‘ฅ) โˆ’(๐‘ฅ+๐‘ฆ) = ๐‘ฅ๐‘ฆ ((๐‘ฆ โˆ’ ๐‘ฅ))/(๐‘ฆ โˆ’ ๐‘ฅ) โˆ’(๐‘ฅ+๐‘ฆ) = ๐‘ฅ๐‘ฆ โˆ’๐‘ฅ โˆ’๐‘ฆ = ๐‘ฅ๐‘ฆ โˆ’๐‘ฅ = ๐‘ฅ๐‘ฆ+๐‘ฆ โˆ’๐‘ฅ = (๐‘ฅ+1) ๐‘ฆ (โˆ’๐‘ฅ)/(๐‘ฅ + 1) = ๐‘ฆ ๐‘ฆ = (โˆ’๐‘ฅ)/(๐‘ฅ + 1) Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘/๐‘‘๐‘ฅ ((โˆ’๐‘ฅ)/(๐‘ฅ + 1)) Using quotient rule As (๐‘ข/๐‘ฃ)โ€ฒ = (๐‘ข^โ€ฒ ๐‘ฃ โˆ’ ๐‘ฃ^โ€ฒ ๐‘ข)/๐‘ฃ^2 where u = โˆ’x & V = x + 1 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘(โˆ’๐‘ฅ)/๐‘‘๐‘ฅ (๐‘ฅ + 1) โˆ’ ๐‘‘(๐‘ฅ + 1)/๐‘‘๐‘ฅ. (โˆ’๐‘ฅ))/(๐‘ฅ + 1)^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’1 (๐‘ฅ + 1) + (1 + 0) ๐‘ฅ)/(๐‘ฅ + 1)^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’๐‘ฅ โˆ’ 1 + ๐‘ฅ)/(๐‘ฅ + 1)^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’1)/(๐‘ฅ + 1)^2 Hence, ๐’…๐’š/๐’…๐’™ = (โˆ’๐Ÿ)/(๐’™ + ๐Ÿ)^๐Ÿ

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.