Ex 5.5, 18 - Show that d/dx (u.v.w)=du/dx v.w + u dv/dx w - Logarithmic Differentiation - Type 1

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.5, 18 If ๐‘ข , ๐‘ฃ and ๐‘ค are functions of ๐‘ฅ, then show that ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ (๐‘ข . ๐‘ฃ . ๐‘ค ) = ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘ฃ. ๐‘ค+๐‘ข . ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ค+๐‘ข . ๐‘ฃ ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ in two ways โˆ’ first by repeated application of product rule, second by logarithmic differentiation. By product Rule Let ๐‘ฆ=๐‘ข๐‘ฃ๐‘ค Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘ข ๐‘ฃ ๐‘ค๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ ๐‘ข๐‘ฃ๏ทฏ ๐‘ค๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ= ๐‘‘ ๐‘ข๐‘ฃ๏ทฏ ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ค + ๐‘‘ ๐‘ค๏ทฏ ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ข๐‘ฃ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ= ๐‘‘ ๐‘ข๏ทฏ ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ฃ+ ๐‘‘ ๐‘ฃ๏ทฏ ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘ข๏ทฏ๐‘ค + ๐‘‘ ๐‘ค๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ข๐‘ฃ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ฃ.๐‘ค+ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ข.๐‘ค + ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ข๐‘ฃ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ฃ.๐‘ค+๐‘ข ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ .๐‘ค+๐‘ข.๐‘ฃ. ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ Hence , ๐’…(๐’– . ๐’— . ๐’˜)๏ทฎ๐’…๐’™๏ทฏ = ๐’…๐’–๏ทฎ๐’…๐’™๏ทฏ . ๐’—.๐’˜+๐’– ๐’…๐’—๏ทฎ๐’…๐’™๏ทฏ .๐’˜+๐’–.๐’—. ๐’…๐’˜๏ทฎ๐’…๐’™๏ทฏ Using logarithmic differentiation. Let ๐‘ฆ=๐‘ข๐‘ฃ๐‘ค Taking log both sides log ๐‘ฆ = log ๐‘ข๐‘ฃ๐‘ค๏ทฏ log ๐‘ฆ=log ๐‘ข+ log ๏ทฎ๐‘ฃ๏ทฏ+ log๏ทฎ๐‘ค๏ทฏ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘ log๏ทฎ๐‘ฆ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ log ๐‘ข + log ๏ทฎ๐‘ฃ๏ทฏ + log๏ทฎ๐‘ค๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘ log๏ทฎ๐‘ฆ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฆ๏ทฏ = ๐‘‘ log ๐‘ข๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ log ๏ทฎ๐‘ฃ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ log๏ทฎ๐‘ค๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘ log๏ทฎ๐‘ฆ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฆ๏ทฏ . ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ log ๐‘ข๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ log ๏ทฎ๐‘ฃ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ log๏ทฎ๐‘ค๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ 1๏ทฎ๐‘ฆ๏ทฏ . ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 1๏ทฎ๐‘ข๏ทฏ . ๐‘‘ ๐‘ข๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 1๏ทฎ๐‘ฃ๏ทฏ. ๐‘‘ ๐‘ฃ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 1๏ทฎ๐‘ค๏ทฏ . ๐‘‘ ๐‘ค๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ฆ 1๏ทฎ๐‘ข๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 1๏ทฎ๐‘ฃ๏ทฏ. ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 1๏ทฎ๐‘ค๏ทฏ . ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ข๐‘ฃ๐‘ค 1๏ทฎ๐‘ข๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 1๏ทฎ๐‘ฃ๏ทฏ. ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 1๏ทฎ๐‘ค๏ทฏ . ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ข๐‘ฃ๐‘ค . 1๏ทฎ๐‘ข๏ทฏ . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘ข๐‘ฃ๐‘ค . 1๏ทฎ๐‘ฃ๏ทฏ. ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘ข๐‘ฃ๐‘ค . 1๏ทฎ๐‘ค๏ทฏ. ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ฃ๐‘ค . ๐‘‘๐‘ข๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘ข๐‘ค . ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘ข๐‘ฃ . ๐‘‘๐‘ค๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐’…(๐’– . ๐’— . ๐’˜)๏ทฎ๐’…๐’™๏ทฏ = ๐’…๐’–๏ทฎ๐’…๐’™๏ทฏ . ๐’—๐’˜+๐’– . ๐’…๐’—๏ทฎ๐’…๐’™๏ทฏ .๐’˜+๐’–.๐’—. ๐’…๐’˜๏ทฎ๐’…๐’™๏ทฏ

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CA Maninder Singh
CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .
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