1. Chapter 5 Class 12 Continuity and Differentiability
2. Serial order wise
3. Ex 5.5

Transcript

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 โ 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.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.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

Ex 5.5