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Ex 5.7, 9 - Find second order derivatives of log (log x)

Ex 5.7, 9 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.7, 9 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Transcript

Ex 5.7, 9 Find the second order derivatives of the function 〖 log〗⁡〖 (log⁡〖𝑥)〗 〗 Let y =〖 log〗⁡〖 (log⁡〖𝑥)〗 〗 Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦/𝑑𝑥 = (𝑑(〖 log〗⁡〖 (log⁡〖𝑥)〗 〗))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 1/log⁡𝑥 . (𝑑(log⁡𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 1/log⁡𝑥 . 1/𝑥 𝑑𝑦/𝑑𝑥 = 1/〖𝑥 . log〗⁡𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 (1/〖𝑥 . log〗⁡𝑥 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = ((𝑑(1))/𝑑𝑥 (〖𝑥 . log〗⁡𝑥 ) − (𝑑 (〖𝑥 . log〗⁡𝑥 ))/𝑑𝑥 . 1 )/(〖𝑥 . log〗⁡𝑥 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (0 . (〖𝑥 . log〗⁡𝑥 ) − (𝑑 (〖𝑥 . log〗⁡𝑥 ))/𝑑𝑥 . 1 )/(〖𝑥 . log〗⁡𝑥 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (− (𝑑 (〖𝑥 . log〗⁡𝑥 ))/𝑑𝑥)/(〖𝑥 . log〗⁡𝑥 )^2 using Quotient Rule As, (𝑢/𝑣)^′= (𝑢’𝑣 − 𝑣’𝑢)/𝑣^2 where u = 1 & v = x log x (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−[(𝑑(𝑥))/𝑑𝑥 .log⁡𝑥 + (𝑑(log⁡〖𝑥)〗)/𝑑𝑥 . 𝑥])/(〖𝑥 . log〗⁡𝑥 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = [1.log⁡〖𝑥 + 1/𝑥 × 𝑥〗 ]/( (𝑥.log⁡𝑥 )^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (− [log⁡𝑥 +1])/( (〖𝑥 . log〗⁡𝑥 )^2 ) Thus, (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (− [𝒍𝒐𝒈⁡𝒙 +𝟏])/( (〖𝒙 . 𝒍𝒐𝒈〗⁡𝒙 )^𝟐 ) using product Rule in 〖𝑥. 𝑙𝑜𝑔〗⁡𝑥 (uv’) = u’v + uv’

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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.