Ex 5.7

Chapter 5 Class 12 Continuity and Differentiability
Serial order wise

### 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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.