Ex 5.7, 10 - Find second order derivatives of sin (log x)

Ex 5.7, 10 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.7, 10 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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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⁡(log⁡𝑥) . (𝑑(log⁡〖𝑥)〗)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = cos⁡(log⁡𝑥) . 1/𝑥 𝑑𝑦/𝑑𝑥 = (cos⁡(log⁡𝑥))/𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 ((cos⁡(log⁡𝑥))/𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑/𝑑𝑥 ((cos⁡(log⁡𝑥))/𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = ((𝑑(cos⁡(log⁡𝑥)))/𝑑𝑥 . 𝑥 − (𝑑 (𝑥))/𝑑𝑥 . cos⁡(log⁡𝑥))/𝑥^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗⁡(log⁡𝑥 ) . 𝑑(log⁡𝑥 )/𝑑𝑥 . 𝑥 − 1 . cos⁡(log⁡𝑥))/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 〗⁡(log⁡𝑥 ) . 1/𝑥 . 𝑥 − cos ⁡(log⁡𝑥))/𝑥^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−〖sin 〗⁡(log⁡𝑥 ) − cos⁡(log⁡𝑥))/𝑥^2 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (− [〖𝒔𝒊𝒏 〗⁡(𝒍𝒐𝒈⁡𝒙 ) + 𝒄𝒐𝒔⁡(𝒍𝒐𝒈⁡𝒙)])/𝒙^𝟐

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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 and Computer Science at Teachoo