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Derivative of cosec-1 x (Cosec inverse x) - Teachoo [with Video]

Derivative of cosec-1 x (Cosec inverse x) - Part 2

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Derivative of 〖𝒄𝒐𝒔𝒆𝒄〗^(βˆ’πŸ) 𝒙 𝑓 (π‘₯)=γ€–π‘π‘œπ‘ π‘’π‘γ€—^(βˆ’1) π‘₯ Let π’š= 〖𝒄𝒐𝒔𝒆𝒄〗^(βˆ’πŸ) 𝒙 cosec⁑〖𝑦=π‘₯γ€— 𝒙=πœπ¨π¬πžπœβ‘γ€–π’š γ€— Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑π‘₯/𝑑π‘₯ = (𝑑 (cosec⁑𝑦 ))/𝑑π‘₯ 1 = (𝑑 (cosec⁑𝑦 ))/𝑑π‘₯ Γ— 𝑑𝑦/𝑑𝑦 1 = (𝑑 (cosec⁑𝑦 ))/𝑑𝑦 Γ— 𝑑𝑦/𝑑π‘₯ 1 = βˆ’cosec⁑𝑦 .cot⁑𝑦 . 𝑑𝑦/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 1/(γ€–βˆ’cosec〗⁑𝑦 .π’„π’π’•β‘π’š ) 𝑑𝑦/𝑑π‘₯ = 1/(γ€–βˆ’cosec〗⁑𝑦 . √(γ€–πœπ¨π’”π’†π’„γ€—^πŸβ‘π’š βˆ’ 𝟏)) Putting value of π‘π‘œπ‘ π‘’π‘β‘π‘¦ = π‘₯ 𝑑𝑦/𝑑π‘₯ = (βˆ’1)/(π‘₯ √(π‘₯^2 βˆ’ 1 ) ) Hence 𝒅(〖𝒄𝒐𝒔𝒆𝒄〗^(β€“πŸ) 𝒙)/𝒅𝒙 = (βˆ’πŸ)/(𝒙 √(𝒙^𝟐 βˆ’ 𝟏 ) ) As cot2 ΞΈ = cosec2 ΞΈ – 1, cot ΞΈ = √("cosec2 ΞΈ – 1" ) As 𝑦 = co〖𝑠𝑒𝑐〗^(βˆ’1) π‘₯ So, coπ’”π’†π’„β‘π’š = 𝒙

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Davneet Singh

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.