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Example 41 Find f β€²(x) if 𝑓 (π‘₯) = (sin⁑π‘₯ )^(sin π‘₯)⁑ for all 0 < x < Ο€.Let 〖𝑦=(sin⁑π‘₯ )γ€—^(sin π‘₯) Taking log on both sides log 𝑦=log (γ€–sin⁑π‘₯γ€—^sin⁑π‘₯ ) π₯𝐨𝐠 π’š=π’”π’Šπ’β‘π’™ . π₯𝐨𝐠 (π’”π’Šπ’β‘π’™ ) Differentiating both sides 𝑀.π‘Ÿ.𝑑. x 𝑑(log⁑𝑦)/𝑑π‘₯ = 𝑑(sin⁑π‘₯ . log (sin⁑π‘₯ ))/𝑑π‘₯ 𝑑(log⁑𝑦 )/𝑑π‘₯ (𝑑𝑦/𝑑𝑦) = 𝑑(sin⁑π‘₯ . log (sin⁑π‘₯ ))/𝑑π‘₯ (As π‘™π‘œπ‘”β‘(π‘Ž^𝑏 )=𝑏 . π‘™π‘œπ‘”β‘π‘Ž) 𝑑(log⁑𝑦 )/𝑑𝑦 (𝑑𝑦/𝑑π‘₯) = 𝑑(sin⁑π‘₯ . log (sin⁑π‘₯ ))/𝑑π‘₯ 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = 𝑑(sin⁑π‘₯ . log (sin⁑π‘₯ ))/𝑑π‘₯ 1/𝑦 𝑑𝑦/𝑑π‘₯ = 𝑑(sin⁑π‘₯ )/𝑑π‘₯ . γ€– log 〗⁑(sin⁑π‘₯ ) + 𝑑(γ€–log 〗⁑(sin⁑π‘₯ ) )/𝑑π‘₯ . sin⁑π‘₯ 1/𝑦 𝑑𝑦/𝑑π‘₯ = cos⁑π‘₯ . γ€–log 〗⁑(sin⁑π‘₯ ) + 1/sin⁑π‘₯ . 𝑑(sin⁑π‘₯ )/𝑑π‘₯ . sin⁑π‘₯ 1/𝑦 𝑑𝑦/𝑑π‘₯ = γ€–cos 〗⁑. γ€–log 〗⁑(sin⁑π‘₯ ) + 1/sin⁑π‘₯ . cos⁑π‘₯.sin π‘₯ 1/𝑦 𝑑𝑦/𝑑π‘₯ = cos⁑π‘₯.log sin π‘₯+cos⁑π‘₯ Using Product rule (uv)’ = u’v + v’u where u = sin x & v = log (sin x) 𝑑𝑦/𝑑π‘₯ = 𝑦 (cos⁑π‘₯.log sin π‘₯+cos⁑π‘₯ ) Putting value of y = (𝑠𝑖𝑛⁑π‘₯ )^𝑠𝑖𝑛⁑π‘₯ 𝑑𝑦/𝑑π‘₯ = (sin⁑π‘₯ )^sin⁑π‘₯ (cos⁑π‘₯.log sin π‘₯+cos⁑π‘₯ ) 𝑑𝑦/𝑑π‘₯ = (sin⁑π‘₯ )^sin⁑π‘₯ . cos⁑π‘₯ (log sin π‘₯+1) π’…π’š/𝒅𝒙 = γ€–(𝟏+π₯𝐨𝐠 (𝐬𝐒𝐧 𝒙)) (π’”π’Šπ’β‘π’™ )γ€—^π’”π’Šπ’β‘π’™ . 𝒄𝒐𝒔⁑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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo