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Ex 5.7, 12 - If y = cos^-1 x, Find d^2y/dx^2 in terms of y alone

Ex 5.7, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.7, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Ex 5.7, 12 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

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Transcript

Ex 5.7, 12 If y= 〖𝑐𝑜𝑠〗^(−1) 𝑥 , Find 𝑑2𝑦/𝑑𝑥2 in terms of 𝑦 alone.Let y = 〖𝑐𝑜𝑠〗^(−1) 𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦/𝑑𝑥 = (𝑑(〖𝑐𝑜𝑠〗^(−1) 𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = (−1)/√(1 − 𝑥^2 ) Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 ((−1)/√(1 − 𝑥^2 )) ("As" 𝑑(〖𝑐𝑜𝑠〗^(−1) 𝑥)/𝑑𝑥=(−1)/√(1 − 𝑥^2 )) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑/𝑑𝑥 ((−1)/√(1 − 𝑥^2 )) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −𝑑/𝑑𝑥 (1−𝑥^2 )^((−1)/2) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −((−1)/2) (1−𝑥^2 )^((−1)/2 − 1)× (1−𝑥^2 )^′ (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 1/2 (1−𝑥^2 )^((−3)/2) × (−2𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −𝑥(1−𝑥^2 )^((−3)/2) (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (− 𝒙)/( (𝟏 − 𝒙^𝟐 )^(𝟑/𝟐 ) ) But we need to calculate (𝑑^2 𝑦)/(𝑑𝑥^2 ) in terms of y . y = 〖𝑐𝑜𝑠〗^(−1) 𝑥 cos⁡𝑦 = 𝑥 𝑥 = cos⁡𝑦 Putting 𝑥 = cos⁡𝑦 in equation (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (− 𝑥)/( (1 − 𝑥^2 )^(3/2 ) ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (−cos⁡𝑦 " " )/( (1 − 〖(cos⁡𝑦)〗^2 )^(3/2 ) ) = (− cos⁡𝑦 " " )/( 〖(sin2⁡〖𝑦) 〗〗^(3/2 ) ) (As 1−cos⁡〖2 〗 𝜃 = sin⁡2 𝜃) = (− cos⁡𝑦 " " )/( 〖(sin⁡〖𝑦) 〗〗^(2 × 3/2 ) ) = (− 𝑐𝑜𝑠⁡𝑦)/( 〖(𝑠𝑖𝑛⁡〖𝑦) 〗〗^3 ) = (− 𝑐𝑜𝑠⁡𝑦)/sin⁡𝑦 ×1/( 〖(𝑠𝑖𝑛⁡𝑦)〗^2 ) = −𝐜𝐨𝐭⁡𝒚 𝒄𝒐𝒔𝒆𝒄^𝟐 𝒚

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.