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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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.