If y = Determinant | f(x) g(x) h(x) l m n a b c|, prove dy/dx = | f'(x

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Misc  22 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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Misc  22 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

Misc  22 - Chapter 5 Class 12 Continuity and Differentiability - Part 4 Misc  22 - Chapter 5 Class 12 Continuity and Differentiability - Part 5

  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Concept wise

Transcript

Misc 22 (Method 1) If ๐‘ฆ = |โ–ˆ( ๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ Here ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| Expanding determinant ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |๐‘“โ€ฒ(๐‘ฅ)| |โ– 8(๐‘š&๐‘›@๐‘&๐‘)||โˆ’๐‘”โ€ฒ(๐‘ฅ) | |โ– 8(๐‘™&๐‘›@๐‘Ž&๐‘)||1+ โ„Žโ€ฒ(๐‘ฅ) ||โ– 8(๐‘™&๐‘š@๐‘Ž&๐‘)| ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘“โ€ฒ(๐‘ฅ) (๐‘š๐‘ โˆ’๐‘๐‘›)โˆ’๐‘”โ€ฒ(๐‘›) (๐‘™๐‘โˆ’๐‘Ž๐‘›) + โ„Žโ€ฒ(๐‘›) (๐‘™๐‘โˆ’๐‘Ž๐‘š) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘š๐‘ โˆ’๐‘๐‘›) ๐‘“โ€ฒ(๐‘ฅ)โˆ’(๐‘™๐‘โˆ’๐‘Ž๐‘›)๐‘”โ€ฒ(๐‘ฅ) +(๐‘™๐‘โˆ’๐‘Ž๐‘š) โ„Žโ€ฒ(๐‘ฅ) Hence We need to prove that ๐’…๐’š/๐’…๐’™ = (๐‘š๐‘ โˆ’๐‘๐‘›) ๐‘“โ€ฒ(๐‘ฅ)โˆ’(๐‘™๐‘โˆ’๐‘Ž๐‘›)๐‘”โ€ฒ(๐‘ฅ) +(๐‘™๐‘โˆ’๐‘Ž๐‘š) โ„Žโ€ฒ(๐‘ฅ) Now, ๐‘ฆ = |โ–ˆ( ๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| Expanding determinant ๐‘ฆ = ๐‘“(๐‘ฅ)|โ– 8(๐‘š&๐‘›@๐‘&๐‘)|โˆ’ ๐‘”(๐‘ฅ)|โ– 8(๐‘™&๐‘›@๐‘Ž&๐‘)|+ โ„Ž(๐‘ฅ)|โ– 8(๐‘™&๐‘š@๐‘Ž&๐‘)| ๐‘ฆ = ๐‘“(๐‘ฅ) (๐‘š๐‘ โˆ’๐‘๐‘›)โˆ’๐‘”(๐‘›) (๐‘™๐‘โˆ’๐‘Ž๐‘›) + โ„Ž(๐‘›) (๐‘™๐‘โˆ’๐‘Ž๐‘š) ๐‘ฆ = (๐‘š๐‘ โˆ’๐‘๐‘›) ๐‘“(๐‘ฅ)โˆ’(๐‘™๐‘โˆ’๐‘Ž๐‘›)๐‘”(๐‘ฅ)" +" (๐‘™๐‘โˆ’๐‘Ž๐‘š) โ„Ž(๐‘ฅ)" " Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘((๐‘š๐‘ โˆ’ ๐‘๐‘›) ๐‘“(๐‘ฅ) โˆ’ (๐‘™๐‘ โˆ’ ๐‘Ž๐‘›)๐‘”(๐‘ฅ)" +" (๐‘™๐‘ โˆ’ ๐‘Ž๐‘š) โ„Ž(๐‘ฅ)" " )/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘((๐‘š๐‘ โˆ’ ๐‘๐‘›) ๐‘“(๐‘ฅ))/๐‘‘๐‘ฅ โˆ’ ๐‘‘((๐‘™๐‘ โˆ’ ๐‘Ž๐‘›)๐‘”(๐‘ฅ))/๐‘‘๐‘ฅ + ๐‘‘((๐‘™๐‘ โˆ’ ๐‘Ž๐‘š) โ„Ž(๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘š๐‘โˆ’๐‘๐‘›) ๐‘‘(๐‘“(๐‘ฅ))/๐‘‘๐‘ฅ โˆ’ (๐‘™๐‘โˆ’๐‘Ž๐‘›) ๐‘‘(๐‘”(๐‘ฅ))/๐‘‘๐‘ฅ + (๐‘™๐‘โˆ’๐‘Ž๐‘š) ๐‘‘(โ„Ž(๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘š๐‘โˆ’๐‘๐‘›) ๐‘“โ€ฒ(๐‘ฅ)โˆ’(๐‘™๐‘โˆ’๐‘Ž๐‘›) ๐‘”โ€ฒ(๐‘ฅ) + (๐‘™๐‘โˆ’๐‘Ž๐‘š) โ„Žโ€ฒ(๐‘ฅ)" " Hence proved Misc 22 (Method 2) If ๐‘ฆ = |โ–ˆ( ๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| , prove that ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| To Differentiate a determinant, We differentiate one row (or one column) at a time keeping others unchanged If ๐‘ฆ = |โ–ˆ( ๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| + |โ–ˆ(๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@(๐‘™)^โ€ฒ (๐‘š)^โ€ฒ (๐‘›)^โ€ฒ@๐‘Ž ๐‘ ๐‘ )| + |โ–ˆ( ๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@(๐‘Ž)โ€ฒ (๐‘)โ€ฒ (๐‘)โ€ฒ )| ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| + |โ–ˆ(๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@0 0 0 @๐‘Ž ๐‘ ๐‘ )| + |โ–ˆ( ๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) โ„Ž(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@0 0 0 )| ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| + 0 + 0 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )| Hence proved. Using property If any one Row or column is 0 , then value of determinate is also 0 ๐‘ )| , prove that ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = |โ–ˆ( ๐‘“โ€ฒ(๐‘ฅ) ๐‘”โ€ฒ(๐‘ฅ) โ„Žโ€ฒ(๐‘ฅ)@๐‘™ ๐‘š ๐‘›@๐‘Ž ๐‘ ๐‘ )|

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.