Ex 5.6, 8 - Find dy/dx, x = a (cos t + log tan t/2) - Ex 5.6

Ex 5.6, 8 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.6, 8 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Ex 5.6, 8 - Chapter 5 Class 12 Continuity and Differentiability - Part 4 Ex 5.6, 8 - Chapter 5 Class 12 Continuity and Differentiability - Part 5

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 5.6, 8 If x and y are connected parametrically by the equations without eliminating the parameter, Find 𝑑𝑦/𝑑π‘₯, π‘₯ = π‘Ž(cos⁑〖𝑑+log⁑〖 tan〗⁑〖𝑑/2γ€— γ€— ) , 𝑦=π‘Ž sin⁑𝑑Here 𝑑𝑦/𝑑π‘₯ = (𝑑𝑦/𝑑𝑑)/(𝑑π‘₯/𝑑𝑑) Calculating π’…π’š/𝒅𝒕 𝑦 = π‘Ž sin⁑𝑑 𝑑𝑦/𝑑𝑑 " " = 𝑑(π‘Ž sin⁑𝑑)/𝑑𝑑 𝑑𝑦/𝑑𝑑 " " = π‘Ž 𝑑( sin⁑𝑑)/𝑑𝑑 𝑑𝑦/𝑑𝑑 " " = π‘Ž cos⁑𝑑 Calculating 𝒅𝒙/𝒅𝒕 π‘₯ = π‘Ž(cos⁑〖𝑑+log⁑〖 tan〗⁑〖𝑑/2γ€— γ€— ) 𝑑π‘₯/𝑑𝑑 = 𝑑(π‘Ž(cos⁑〖𝑑+log⁑〖 tan 〗⁑〖𝑑/2γ€— γ€— ))/𝑑𝑑 𝑑π‘₯/𝑑𝑑 = a ((𝑑(cos⁑𝑑 ) )/𝑑𝑑 +𝑑(log⁑(γ€–tan 〗⁑〖𝑑/2γ€— ) )/𝑑𝑑) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +𝑑(log⁑(tan⁑〖 𝑑/2γ€— ) )/𝑑𝑑) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +1/tan⁑〖 𝑑/2γ€— . 𝑑(γ€–tan 〗⁑〖𝑑/2γ€— )/𝑑𝑑) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +1/tan⁑〖 𝑑/2γ€— . (sec^2⁑〖𝑑/2γ€— ).𝑑(𝑑/2)/𝑑𝑑 ) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +1/tan⁑〖 𝑑/2γ€— . sec^2 (𝑑/2). 1/2 ) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +cot⁑〖𝑑/2γ€— . 1/(cos^2 𝑑/2) . 1/2 ) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +γ€–cos 〗⁑〖𝑑/2γ€—/sin⁑〖 𝑑/2γ€— . 1/(cos^2 𝑑/2) . 1/2 ) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +1/(sin⁑〖 𝑑/2γ€— cos⁑〖 𝑑/2γ€— ) . 1/2 ) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +1/(2 sin⁑〖𝑑/2γ€— cos⁑〖𝑑/2γ€— )) 𝑑π‘₯/𝑑𝑑 = a (βˆ’sin⁑𝑑 +1/sin⁑𝑑 ) 𝑑π‘₯/𝑑𝑑 = a ((βˆ’sin^2⁑𝑑 + 1)/sin⁑𝑑 ) 𝑑π‘₯/𝑑𝑑 = a ((1 βˆ’ sin^2⁑𝑑)/sin⁑𝑑 ) 𝒅𝒙/𝒅𝒕 = a (〖𝒄𝒐𝒔〗^πŸβ‘π’•/π’”π’Šπ’β‘π’• ) We know that sin 2ΞΈ =2sin ΞΈ cos ΞΈ Replacing ΞΈ by 𝑑/2 sin t = 2 sin 𝑑/2 . cos 𝑑/2 Therefore, π’…π’š/𝒅𝒙 " =" (π’…π’š/𝒅𝒕)/(𝒅𝒙/𝒅𝒕) 𝑑𝑦/𝑑π‘₯ = (π‘Ž cos⁑𝑑)/(γ€–π‘Ž cos^2〗⁑𝑑/sin⁑𝑑 ) 𝑑𝑦/𝑑π‘₯ = π‘Ž cos⁑𝑑 Γ— sin⁑𝑑/(π‘Ž cos^2 𝑑) 𝑑𝑦/𝑑π‘₯ = (π‘Ž cos⁑𝑑 . sin⁑𝑑)/(π‘Ž cos⁑𝑑 .cos⁑𝑑 ) 𝑑𝑦/𝑑π‘₯ = sin⁑𝑑/cos⁑𝑑 π’…π’š/𝒅𝒙 = π­πšπ§β‘π’•

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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.