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Misc 9 - Differentiate (sin x - cos x)^*(sin x - cos x) - Chapter 5

Misc  9 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Misc  9 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Misc 9 Differentiate w.r.t. x the function, (sin⁑π‘₯βˆ’cos⁑π‘₯ )^((sin⁑〖π‘₯βˆ’cos⁑〖π‘₯)γ€— γ€— ), πœ‹/4 <π‘₯< 3πœ‹/4 Let y = (sin⁑π‘₯βˆ’cos⁑π‘₯ )^((sin⁑〖π‘₯βˆ’cos⁑〖π‘₯)γ€— γ€— ) Taking log on both sides log⁑𝑦 = log (sin⁑π‘₯βˆ’cos⁑π‘₯ )^((sin⁑〖π‘₯βˆ’cos⁑〖π‘₯)γ€— γ€— ) log⁑𝑦 = (sin⁑π‘₯βˆ’cos⁑π‘₯ ). γ€– log〗⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑(log⁑𝑦 )/𝑑π‘₯ = 𝑑((sin⁑π‘₯ βˆ’ cos⁑π‘₯ ). γ€– log〗⁑(sin⁑π‘₯ βˆ’ cos⁑π‘₯ ) )/𝑑π‘₯ 𝑑(log⁑𝑦 )/𝑑π‘₯ (𝑑𝑦/𝑑𝑦) = 𝑑((sin⁑π‘₯βˆ’cos⁑π‘₯ ). γ€– log〗⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— )/𝑑π‘₯ 𝑑(log⁑𝑦 )/𝑑𝑦 (𝑑𝑦/𝑑π‘₯) = 𝑑((sin⁑π‘₯βˆ’cos⁑π‘₯ ). γ€– log〗⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— )/𝑑π‘₯ 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = 𝑑((sin⁑π‘₯βˆ’cos⁑π‘₯ ). γ€– log〗⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— " " )/𝑑π‘₯ " " 1/𝑦. 𝑑𝑦/𝑑π‘₯ = 𝑑(sin⁑π‘₯ βˆ’ cos⁑π‘₯ )/𝑑π‘₯ . γ€– log 〗⁑(sin⁑π‘₯βˆ’cos⁑π‘₯ ) + 𝑑(γ€– log〗⁑〖 (sin⁑π‘₯ βˆ’ cos⁑π‘₯ )γ€— )/𝑑π‘₯ .(sin⁑π‘₯βˆ’cos⁑π‘₯ ) 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = (cos⁑π‘₯βˆ’(βˆ’sin⁑π‘₯ )). log⁑〖 (sin⁑π‘₯βˆ’γ€– cos〗⁑π‘₯ )γ€— + 1/((sin⁑π‘₯ βˆ’ cos⁑π‘₯ ) ) . 𝑑(sin⁑π‘₯ βˆ’ cos⁑π‘₯ )/𝑑π‘₯ . (sin⁑π‘₯βˆ’γ€– cos〗⁑π‘₯ ) 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = (cos⁑π‘₯+sin⁑π‘₯ ) . log⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— + 1/((sin⁑π‘₯βˆ’cos⁑π‘₯ ) ) . (cos⁑π‘₯βˆ’(βˆ’sin⁑π‘₯ )) . (sin⁑π‘₯βˆ’cos⁑π‘₯ ) Using product rule (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 where u = sin x βˆ’ cos x & v = log (sin x βˆ’ cos x) 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = (cos⁑π‘₯+sin⁑π‘₯ ) . log⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— + 1/((sin⁑π‘₯βˆ’cos⁑π‘₯ ) ) . (cos⁑π‘₯+sin⁑π‘₯ ) . (sin⁑π‘₯βˆ’cos⁑π‘₯ ) 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = (cos⁑π‘₯+sin⁑π‘₯ ) . log⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )γ€— + (cos⁑π‘₯+sin⁑π‘₯ ) 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = (cos⁑π‘₯+sin⁑π‘₯ ) . (log⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )+1γ€— ) 𝑑𝑦/𝑑π‘₯ = 𝑦(cos⁑π‘₯+sin⁑π‘₯ ) . (log⁑〖 (sin⁑π‘₯βˆ’cos⁑π‘₯ )+1γ€— ) π’…π’š/𝒅𝒙 = (π’”π’Šπ’β‘π’™βˆ’π’„π’π’”β‘π’™ )^((π’”π’Šπ’β‘γ€–π’™βˆ’π’„π’π’”β‘γ€–π’™)γ€— γ€— ) (𝒄𝒐𝒔⁑𝒙+π’”π’Šπ’β‘π’™ )(π’π’π’ˆβ‘γ€– (π’”π’Šπ’β‘π’™βˆ’π’„π’π’”β‘π’™ )+πŸγ€— )

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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.