# Ex 5.7, 10 - Chapter 5 Class 12 Continuity and Differentiability (Term 1)

Last updated at March 11, 2021 by Teachoo

Last updated at March 11, 2021 by Teachoo

Transcript

Ex 5.7, 10 Find the second order derivatives of the function 〖 sin〗〖 (log〖𝑥)〗 〗 Let y = 〖 sin〗〖 (log〖𝑥)〗 〗 Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦/𝑑𝑥 = (𝑑(〖 sin〗〖 (log〖𝑥)〗 〗))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = cos(log𝑥) . (𝑑(log〖𝑥)〗)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = cos(log𝑥) . 1/𝑥 𝑑𝑦/𝑑𝑥 = (cos(log𝑥))/𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 ((cos(log𝑥))/𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑/𝑑𝑥 ((cos(log𝑥))/𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = ((𝑑(cos(log𝑥)))/𝑑𝑥 . 𝑥 − (𝑑 (𝑥))/𝑑𝑥 . cos(log𝑥))/𝑥^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗(log𝑥 ) . 𝑑(log𝑥 )/𝑑𝑥 . 𝑥 − 1 . cos(log𝑥))/𝑥^2 Using Quotient Rule As, (𝑢/𝑣)^′= (𝑢’𝑣 − 𝑣’𝑢)/𝑣^2 where v = cos (log x) & v = x (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗(log𝑥 ) . 1/𝑥 . 𝑥 − cos (log𝑥))/𝑥^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗(log𝑥 ) − cos(log𝑥))/𝑥^2 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (− [〖𝒔𝒊𝒏 〗(𝒍𝒐𝒈𝒙 ) + 𝒄𝒐𝒔(𝒍𝒐𝒈𝒙)])/𝒙^𝟐

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.