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Example 30 - Differentiate root (x-3) (x^2+4) / 3x^2 + 4x + 5

Example 30 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Example 30 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Example 30 Differentiate √(((π‘₯βˆ’3) (π‘₯2+4))/( 3π‘₯2+ 4π‘₯ + 5)) 𝑀.π‘Ÿ.𝑑. π‘₯.Let y =√(((π‘₯ βˆ’ 3) (π‘₯^2 + 4))/( 3π‘₯2+ 4π‘₯ + 5)) Taking log on both sides π’π’π’ˆβ‘π’š = π’π’π’ˆ √(((𝒙 βˆ’ πŸ‘) (π’™πŸ + πŸ’))/( πŸ‘π’™πŸ+ πŸ’π’™ + πŸ“)) log⁑𝑦 = log⁑ (((π‘₯ βˆ’ 3) (π‘₯2 + 4))/( 3π‘₯2+ 4π‘₯ + 5))^(1/2) π’π’π’ˆβ‘π’š = 𝟏/𝟐 log ((π‘₯ βˆ’ 3) (π‘₯2 + 4))/((3π‘₯2+ 4π‘₯ + 5) ) (π‘ˆπ‘ π‘–π‘›π‘” logβ‘γ€–π‘Ž^𝑏 γ€—=𝑏 logβ‘π‘Ž) π’π’π’ˆβ‘π’š = 𝟏/𝟐 log ((π‘₯ βˆ’ 3) (π‘₯2 + 4))/((3π‘₯2+ 4π‘₯ + 5) ) π‘™π‘œπ‘”β‘π‘¦ = 1/2 (log⁑〖 (π‘₯βˆ’3)γ€—+γ€–log 〗⁑(π‘₯2 + 4)βˆ’log⁑〖 (3π‘₯2 + 4π‘₯ + 5)γ€— ) Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ (𝑑 (log⁑𝑦))/𝑑π‘₯ = 1/2 ((𝑑(log⁑(π‘₯βˆ’3)+γ€–log 〗⁑〖(π‘₯^2+4)βˆ’log⁑〖 (3π‘₯^2+4π‘₯+5)γ€— γ€— ) )/𝑑π‘₯) (𝑑 (log⁑𝑦))/𝑑π‘₯ = 1/2 ((𝑑(log⁑(π‘₯βˆ’3)))/𝑑π‘₯ " + " 𝑑(log⁑(π‘₯^2+4) )/𝑑π‘₯ " βˆ’ " 𝑑(log⁑(3π‘₯^2+4π‘₯+5) )/𝑑π‘₯) 1/𝑦 (𝑑𝑦/𝑑π‘₯) = 1/2 [1/(π‘₯ βˆ’ 3) " . " 𝑑(π‘₯ βˆ’3)/𝑑π‘₯ " + " 1/(π‘₯^2 + 4) " . " (𝑑 (π‘₯^2 + 4))/𝑑π‘₯ " – " 1/((3π‘₯^2+ 4π‘₯+ 5)) " . " (𝑑 (3π‘₯^2+ 4π‘₯+ 5))/(𝑑π‘₯ )] π‘ˆπ‘ π‘–π‘›π‘” π‘™π‘œπ‘”β‘π‘Žπ‘=π‘™π‘œπ‘”β‘π‘Ž+π‘π‘œπ‘ β‘π‘ &π‘™π‘œπ‘”β‘γ€–π‘Ž/𝑏〗=π‘™π‘œπ‘”β‘γ€–π‘Žβˆ’π‘™π‘œπ‘”β‘π‘ γ€— 1/𝑦 (𝑑𝑦/𝑑π‘₯) = 1/2 [1/((π‘₯ βˆ’ 3) ) " " (1βˆ’0) "+ " 1/(π‘₯^2+ 4) " " (2π‘₯ + 0)" βˆ’ " 1/(3π‘₯^2+ 4π‘₯ + 5) " " (6π‘₯ +4+0)" " ] 1/𝑦 (𝑑𝑦/𝑑π‘₯) = 1/2 (1/((π‘₯ βˆ’ 3) )+2π‘₯/(π‘₯^2+ 4)βˆ’(6π‘₯ + 4)/(3π‘₯^2 + 4π‘₯ + 5)) 𝑑𝑦/𝑑π‘₯ = 𝑦/2 (1/((π‘₯ βˆ’ 3) )+2π‘₯/(π‘₯^2+ 4)βˆ’(6π‘₯ + 4)/(3π‘₯^2 + 4π‘₯ + 5)) π’…π’š/𝒅𝒙 = 𝟏/𝟐 √(((𝒙 βˆ’ πŸ‘)(𝒙^𝟐+ πŸ’))/(πŸ‘π’™^𝟐+ πŸ’π’™ + πŸ“)) (𝟏/((𝒙 βˆ’ πŸ‘) )+πŸπ’™/(𝒙^𝟐+ πŸ’)βˆ’(πŸ”π’™ + πŸ’)/(πŸ‘π’™^𝟐 + πŸ’π’™ + πŸ“))

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.