Misc 22  - If y = |f(x) g(x) h(x) l m n a b c|, prove that - Proofs

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Misc 22 (Method 1) If ๐‘ฆ = ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ , prove that ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ Consider ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ Expanding determinate along 1st Row ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ๏ทฏ ๐‘š๏ทฎ๐‘›๏ทฎ๐‘๏ทฎ๐‘๏ทฏ๏ทฏ โˆ’๐‘”โ€ฒ ๐‘ฅ๏ทฏ ๏ทฏ ๐‘™๏ทฎ๐‘›๏ทฎ๐‘Ž๏ทฎ๐‘๏ทฏ๏ทฏ 1+ โ„Žโ€ฒ(๐‘ฅ) ๏ทฏ ๐‘™๏ทฎ๐‘š๏ทฎ๐‘Ž๏ทฎ๐‘๏ทฏ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏโˆ’๐‘”โ€ฒ ๐‘›๏ทฏ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ + โ„Žโ€ฒ(๐‘›) ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏ ๐‘“โ€ฒ ๐‘ฅ๏ทฏโˆ’ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ๐‘”โ€ฒ ๐‘ฅ๏ทฏ + ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ โ„Žโ€ฒ(๐‘ฅ) Hence We need to prove that ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏ ๐‘“โ€ฒ ๐‘ฅ๏ทฏโˆ’ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ๐‘”โ€ฒ ๐‘ฅ๏ทฏ + ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ โ„Žโ€ฒ(๐‘ฅ) Now, ๐‘ฆ = ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ Expanding determinate along 1st Row ๐‘ฆ = ๐‘“ ๐‘ฅ๏ทฏ๏ทฏ ๐‘š๏ทฎ๐‘›๏ทฎ๐‘๏ทฎ๐‘๏ทฏ๏ทฏ โˆ’ ๐‘” ๐‘ฅ๏ทฏ ๏ทฏ ๐‘™๏ทฎ๐‘›๏ทฎ๐‘Ž๏ทฎ๐‘๏ทฏ๏ทฏ + โ„Ž(๐‘ฅ) ๏ทฏ ๐‘™๏ทฎ๐‘š๏ทฎ๐‘Ž๏ทฎ๐‘๏ทฏ๏ทฏ ๐‘ฆ = ๐‘“ ๐‘ฅ๏ทฏ ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏโˆ’๐‘” ๐‘›๏ทฏ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ + โ„Ž(๐‘›) ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ ๐‘ฆ = ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏ ๐‘“ ๐‘ฅ๏ทฏโˆ’ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ๐‘” ๐‘ฅ๏ทฏ + ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ โ„Ž(๐‘ฅ) Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ โˆ’ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ๐‘” ๐‘ฅ๏ทฏ + ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ โ„Ž(๐‘ฅ) ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘š๐‘ โˆ’๐‘๐‘›๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ โˆ’ ๐‘‘ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ๐‘” ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ โ„Ž(๐‘ฅ)๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘š๐‘โˆ’๐‘๐‘›๏ทฏ ๐‘‘ ๐‘“ ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ โˆ’ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ ๐‘‘ ๐‘” ๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ ๐‘‘ โ„Ž(๐‘ฅ)๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘š๐‘โˆ’๐‘๐‘›๏ทฏ ๐‘“โ€ฒ ๐‘ฅ๏ทฏโˆ’ ๐‘™๐‘โˆ’๐‘Ž๐‘›๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ + ๐‘™๐‘โˆ’๐‘Ž๐‘š๏ทฏ โ„Žโ€ฒ(๐‘ฅ) Hence proved Misc 22 (Method 2) If ๐‘ฆ = ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ , prove that ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ To Differentiate a determinant , We differentiate one row (or one column) at a time keeping others unchanged If ๐‘ฆ = ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ + ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ ๐‘™๏ทฏ๏ทฎโ€ฒ๏ทฏ ๐‘š๏ทฏ๏ทฎโ€ฒ๏ทฏ ๐‘›๏ทฏ๏ทฎโ€ฒ๏ทฏ๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ + ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ(๐‘Ž)โ€ฒ (๐‘)โ€ฒ (๐‘)โ€ฒ ๏ทฏ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ + ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ0 0 0 ๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ + ๐‘“ ๐‘ฅ๏ทฏ ๐‘” ๐‘ฅ๏ทฏ โ„Ž(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ0 0 0 ๏ทฏ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ + 0 + 0 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘“โ€ฒ ๐‘ฅ๏ทฏ ๐‘”โ€ฒ ๐‘ฅ๏ทฏ โ„Žโ€ฒ(๐‘ฅ)๏ทฎ๐‘™ ๐‘š ๐‘›๏ทฎ๐‘Ž ๐‘ ๐‘ ๏ทฏ๏ทฏ Hence proved.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.