Example 46 - Find f'(x) if f(x) = (sin x)sin x - NCERT - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 46 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‘₯ ) log 𝑦=sin⁑π‘₯ . log (sin⁑π‘₯ ) Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑(log⁑𝑦)/𝑑π‘₯ = 𝑑(sin⁑π‘₯ . log (sin⁑π‘₯ ))/𝑑π‘₯ 𝑑(log⁑𝑦 )/𝑑π‘₯ (𝑑𝑦/𝑑𝑦) = 𝑑(sin⁑π‘₯ . log (sin⁑π‘₯ ))/𝑑π‘₯ 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‘₯ 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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.