If 𝑦 = sin ^{ -1 } x, then (1-x ^{ 2 } ) y _{ 2 } 𝑖𝑠 equal to
(a) xy _{ 1 }
(b) 𝑥𝑦
(c) xy _{ 2 }
(d) x ^{ 2 }
CBSE Class 12 Sample Paper for 2023 Boards
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CBSE Class 12 Sample Paper for 2023 Boards
Last updated at April 16, 2024 by Teachoo
Question 16 If 𝑦 = 〖𝑠𝑖𝑛〗^(−1) 𝑥, then (1−𝑥^2 ) 𝑦_2 𝑖𝑠 equal to (a) 〖𝑥𝑦〗_1 (b) 𝑥𝑦 (c) 〖𝑥𝑦〗_2 (d) 𝑥^2 Given 𝑦 = 〖𝑠𝑖𝑛〗^(−1) 𝑥 Differentiating w.r.t x 𝑑𝑦/𝑑𝑥=𝑑(sin^(−1)𝑥 )/𝑥 𝒅𝒚/𝒅𝒙=𝟏/√(𝟏 − 𝒙^𝟐 ) 𝑦^′=1/√(1 − 𝑥^2 ) 𝒚^′ × √(𝟏 − 𝒙^𝟐 )=𝟏 Again differentiating w.r.t x (𝑦^′ × √(1 − 𝑥^2 ))^′=𝑑(1)/𝑑𝑥 (𝑦^′ × √(1 − 𝑥^2 ))^′=0 〖𝒚^′′ √(𝟏 − 𝒙^𝟐 )+𝒚^′ (√(𝟏 − 𝒙^𝟐 ))〗^′=𝟎 〖𝑦^′′ √(1 − 𝑥^2 )+𝑦^′ ×1/(2√(1 − 𝑥^2 )) (1 − 𝑥^2 )〗^′=0 𝑦^′′ √(1 − 𝑥^2 )+𝑦^′ ×1/(2√(1 − 𝑥^2 )) × −2𝑥=0 𝑦^′′ √(1 − 𝑥^2 )−𝑦^′ ×𝑥/√(1 − 𝑥^2 ) =0 𝑦^′′ √(1 − 𝑥^2 )=𝑦^′ ×𝑥/√(1 − 𝑥^2 ) 𝑦^′′ √(1 − 𝑥^2 ) ×√(1 − 𝑥^2 )=𝑦^′ 𝑥 𝒚^′′ (𝟏 − 𝒙^𝟐) =𝒚^′ 𝒙 So, the correct answer is (a)