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

Transcript

Ex 5.5, 14 Find 𝑑𝑦/𝑑π‘₯ of the functions in, γ€–(cos⁑〖π‘₯ γ€—)γ€—^𝑦 = γ€–(cos⁑〖𝑦 γ€—)γ€—^π‘₯ Given γ€–(cos⁑π‘₯)γ€—^𝑦 = γ€–(cos⁑𝑦)γ€—^π‘₯ Taking log both sides log γ€–(cos⁑π‘₯)γ€—^𝑦 = log γ€–(cos⁑𝑦)γ€—^π‘₯ 𝑦 . log (cos⁑π‘₯)=π‘₯.log⁑〖(cos⁑𝑦)γ€— Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯. (𝑑(𝑦 . log (cos⁑π‘₯)))/𝑑π‘₯ = 𝑑(π‘₯.γ€– log〗⁑〖(cos⁑𝑦)γ€— )/𝑑π‘₯ (As π‘™π‘œπ‘”β‘(π‘Ž^𝑏 )=𝑏 . π‘™π‘œπ‘”β‘π‘Ž) Finding (𝒅(π’š . π’π’π’ˆ (𝒄𝒐𝒔⁑𝒙)))/𝒅𝒙 (𝑑(𝑦 . π‘™π‘œπ‘” (π‘π‘œπ‘ β‘π‘₯)))/𝑑π‘₯ = (𝑑(𝑦))/𝑑π‘₯ . log co𝑠⁑π‘₯ + (𝑑(π‘™π‘œπ‘” (π‘π‘œπ‘ β‘π‘₯)))/𝑑π‘₯ . 𝑦 = 𝑑𝑦/𝑑π‘₯ . log co𝑠⁑π‘₯ + 1/π‘π‘œπ‘ β‘π‘₯ . 𝑑(π‘π‘œπ‘ β‘π‘₯ )/𝑑π‘₯ . 𝑦 = 𝑑𝑦/𝑑π‘₯ . log co𝑠⁑π‘₯ + 1/π‘π‘œπ‘ β‘π‘₯ . (βˆ’sin⁑π‘₯ ) . 𝑦 = 𝑑𝑦/𝑑π‘₯ . log co𝑠⁑π‘₯ + ((βˆ’sin⁑π‘₯ ))/π‘π‘œπ‘ β‘π‘₯ . 𝑦 = 𝑑𝑦/𝑑π‘₯ . log co𝑠⁑π‘₯βˆ’tan⁑π‘₯. 𝑦 Using product Rule As (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 Finding 𝒅(𝒙.γ€– π’π’π’ˆγ€—β‘γ€–(π’„π’π’”β‘π’š)γ€— )/𝒅𝒙 𝑑(π‘₯.γ€– π‘™π‘œπ‘”γ€—β‘γ€–(π‘π‘œπ‘ β‘π‘¦)γ€— )/𝑑π‘₯ = (𝑑(π‘₯))/𝑑π‘₯ . log co𝑠⁑𝑦 + (𝑑(π‘™π‘œπ‘” (π‘π‘œπ‘ β‘π‘¦)))/𝑑π‘₯ . π‘₯ = log co𝑠⁑𝑦 + 1/π‘π‘œπ‘ β‘π‘¦ . 𝑑(π‘π‘œπ‘ β‘π‘¦ )/𝑑π‘₯ . π‘₯ = log co𝑠⁑𝑦 + 1/π‘π‘œπ‘ β‘π‘¦ . 𝑑(π‘π‘œπ‘ β‘π‘¦ )/𝑑π‘₯ . 𝑑𝑦/𝑑𝑦 . π‘₯ = log co𝑠⁑𝑦 + 1/π‘π‘œπ‘ β‘π‘¦ . 𝑑(π‘π‘œπ‘ β‘π‘¦ )/𝑑𝑦 . 𝑑𝑦/𝑑π‘₯ . π‘₯ Using product Rule As (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 = log co𝑠⁑𝑦 + 1/π‘π‘œπ‘ β‘π‘¦ . (βˆ’sin⁑𝑦) . 𝑑𝑦/𝑑π‘₯ . π‘₯ = log co𝑠⁑𝑦 + βˆ’tan⁑𝑦 . π‘₯ . 𝑑𝑦/𝑑π‘₯ Now , (𝑑(𝑦 . log (cos⁑π‘₯)))/𝑑π‘₯ = 𝑑(π‘₯.γ€– log〗⁑〖(cos⁑𝑦)γ€— )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ log co𝑠⁑π‘₯βˆ’tan⁑π‘₯. 𝑦 = log co𝑠⁑𝑦 βˆ’ tan⁑𝑦 . π‘₯ . 𝑑𝑦/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ log co𝑠⁑π‘₯βˆ’π‘¦ . tan⁑π‘₯ = log co𝑠⁑𝑦 βˆ’ π‘₯ . tan⁑𝑦 . 𝑑𝑦/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ log co𝑠⁑π‘₯+π‘₯ tan 𝑑𝑦/𝑑π‘₯ = log co𝑠⁑𝑦 + 𝑦 tan⁑π‘₯ 𝑑𝑦/𝑑π‘₯ (log co𝑠⁑π‘₯+π‘₯ tan 𝑦) = log co𝑠⁑𝑦 + 𝑦 tan⁑π‘₯ π’…π’š/𝒅𝒙 = (π₯𝐨𝐠 π’„π’π’”β‘π’š " + " π’š 𝒕𝒂𝒏⁑𝒙)/(π₯𝐨𝐠 𝒄𝒐𝒔⁑𝒙 + 𝒙 𝐭𝐚𝐧 π’š)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.