Misc 13 - Chapter 5 Class 12 Continuity and Differentiability

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Misc 13 Find 𝑑𝑦/𝑑𝑥 , if 𝑦=〖𝑠𝑖𝑛〗^(−𝟏) 𝑥+〖𝑠𝑖𝑛〗^(−1) √(1−𝑥2), – 1 ≤ 𝑥 ≤ 1 𝑦=〖𝑠𝑖𝑛〗^(−𝟏) 𝑥+〖𝑠𝑖𝑛〗^(−1) √(1−𝑥^2 ) , – 1 ≤ 𝑥 ≤ 1 Put 𝑥 = 𝑠𝑖𝑛⁡𝜃 𝑦=〖𝑠𝑖𝑛〗^(−𝟏) (sin⁡𝜃)+〖𝑠𝑖𝑛〗^(−1) √(1−sin^2 𝜃 ) 𝑦=𝜃+〖𝑠𝑖𝑛〗^(−1) √(cos^2 𝜃 ) 𝑦=𝜃+〖𝑠𝑖𝑛〗^(−1) (cos 𝜃) 𝑦=𝜃+〖𝑠𝑖𝑛〗^(−1) (sin⁡(𝜋/2 −𝜃) ) 𝑦=𝜃+ (𝜋/2 −𝜃) ("As " 〖𝑠𝑖𝑛〗^(−1) (sin⁡〖θ)〗=θ) (As cos θ = sin (𝜋/2 – θ)) ("As " 〖𝑠𝑖𝑛〗^(−1) (sin⁡〖θ)〗=θ) 𝑦=𝜃−𝜃 + 𝜋/2 𝑦= 𝜋/2 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦/𝑑𝑥 = 𝑑(𝜋/2)/𝑑𝑥 𝒅𝒚/𝒅𝒙 = 0 As derivative of constant is zero, here 𝜋/2 is a constant