Derivative of cosec-1 x (Cosec inverse x)

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Find the derivative of f given by f (x) = cosec–1 𝑥 assuming it exists. Let 𝑦 = cosec^(–1) 𝑥 cosec⁡𝑦= 𝑥 𝑥 =cosec⁡𝑦 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 (𝑑(𝑥))/𝑑𝑥 = (𝑑 (cosec⁡𝑦 ))/𝑑𝑥 1 = (𝑑 (cosec⁡𝑦 ))/𝑑𝑥 We need 𝑑𝑦 in denominator, so multiplying & Dividing by 𝑑𝑦. 1 = (𝑑 (cosec⁡𝑦 ))/𝑑𝑥 × 𝑑𝑦/𝑑𝑦 1 = cosec⁡𝑦 .cot⁡𝑦 . 𝑑𝑦/𝑑𝑥 〖−cosec〗⁡𝑦 .cot⁡𝑦 𝑑𝑦/𝑑𝑥 = 1 𝑑𝑦/𝑑𝑥 = 1/(〖−cosec〗⁡𝑦 .𝒄𝒐𝒕⁡𝒚 ) 𝑑𝑦/𝑑𝑥 = 1/(〖−cosec〗⁡𝑦 . √(〖𝐜𝐨𝒔𝒆𝒄〗^𝟐⁡𝒚−𝟏)) 𝑊𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 cot^2⁡〖𝜃=cosec^2⁡〖𝜃−1〗 〗 cot⁡𝜃= √(cosec^2⁡𝜃−1) Putting value of 𝑐𝑜𝑠𝑒𝑐⁡𝑦 = 𝑥 𝑑𝑦/𝑑𝑥 = 1/(− 𝑥√(𝑥^2 − 1 )) As 𝑦 = cosec^(−1) 𝑥 So, co𝑠𝑒𝑐⁡𝑦 = 𝑥 𝑑𝑦/𝑑𝑥 = (−1)/(𝑥 √(𝑥^2 − 1 ) ) Hence 𝒅(〖𝒄𝒐𝒔𝒆𝒄〗^(–𝟏) 𝒙)/𝒅𝒙 = (−𝟏)/(𝒙 √(𝒙^𝟐 − 𝟏 ) )