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Misc 17 If π‘₯=π‘Ž (cos⁑𝑑 + 𝑑 sin⁑𝑑) and y=π‘Ž (sin⁑𝑑 – 𝑑 cos⁑𝑑), Find (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^We need to find (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 First we find π’…π’š/𝒅𝒙 Here, 𝑑𝑦/𝑑π‘₯ = (𝑑𝑦/𝑑𝑑)/(𝑑π‘₯/𝑑𝑑) Calculating π’…π’š/𝒅𝒕 𝑦=π‘Ž (sin⁑𝑑– 𝑑 cos⁑𝑑 ) Differentiating 𝑀.π‘Ÿ.𝑑. t 𝑑𝑦/𝑑𝑑 = 𝑑(π‘Ž (sin⁑𝑑– 𝑑 cos⁑𝑑 ))/𝑑𝑑 𝑑𝑦/𝑑𝑑 = π‘Ž 𝑑(sin⁑𝑑– 𝑑 cos⁑𝑑 )/𝑑𝑑 𝑑𝑦/𝑑𝑑 = π‘Ž (𝑑(sin⁑𝑑 )/𝑑𝑑 βˆ’ 𝑑(𝑑 cos⁑𝑑 )/𝑑𝑑) 𝑑𝑦/𝑑𝑑 = π‘Ž (cosβ‘π‘‘βˆ’ 𝑑(𝑑 cos⁑𝑑 )/𝑑𝑑) 𝑑𝑦/𝑑𝑑 = π‘Ž (cos⁑𝑑 βˆ’(𝑑𝑑/𝑑𝑑 . cos⁑𝑑+ (𝑑 cos⁑𝑑)/𝑑𝑑 . 𝑑 )) Using Product rule As (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 𝑑𝑦/𝑑𝑑 = π‘Ž (cos⁑𝑑 βˆ’(cos⁑𝑑+(γ€–βˆ’sin〗⁑𝑑 ) . 𝑑)) 𝑑𝑦/𝑑𝑑 = π‘Ž (cos⁑𝑑 βˆ’(cosβ‘π‘‘βˆ’(sin⁑𝑑 ) . 𝑑)) 𝑑𝑦/𝑑𝑑 = π‘Ž (cos⁑𝑑 βˆ’cos⁑𝑑+𝑑 .sin⁑𝑑 ) 𝑑𝑦/𝑑𝑑 = π‘Ž (0+𝑑 sin⁑𝑑 ) π’…π’š/𝒅𝒕 = 𝒂 .𝒕.π’”π’Šπ’β‘π’• Calculating 𝒅𝒙/𝒅𝒕 π‘₯=π‘Ž (cos⁑𝑑+ 𝑑 sin⁑𝑑 ) Differentiating 𝑀.π‘Ÿ.𝑑. t 𝑑π‘₯/𝑑𝑑 = 𝑑(π‘Ž (cos⁑𝑑 + 𝑑 sin⁑𝑑)" " )/𝑑𝑑 𝑑π‘₯/𝑑𝑑 = π‘Ž (𝑑(cos⁑𝑑 + 𝑑 sin⁑𝑑)/𝑑𝑑) 𝑑π‘₯/𝑑𝑑 = π‘Ž (𝑑(cos⁑𝑑)/𝑑𝑑 + 𝑑(𝑑 sin⁑𝑑)/𝑑𝑑) 𝑑π‘₯/𝑑𝑑 = π‘Ž (γ€–βˆ’sin〗⁑𝑑 + 𝑑(𝑑 sin⁑𝑑 )/𝑑𝑑) Using product rule As (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 𝑑π‘₯/𝑑𝑑 = π‘Ž (γ€–βˆ’sin〗⁑𝑑+(𝑑𝑑/𝑑𝑑 . sin⁑𝑑+ 𝑑(sin⁑𝑑 )/𝑑𝑑 . 𝑑 )) 𝑑π‘₯/𝑑𝑑 = π‘Ž (γ€–βˆ’sin〗⁑𝑑+(sin⁑𝑑+cos⁑𝑑 . 𝑑)) 𝑑π‘₯/𝑑𝑑= π‘Ž (βˆ’sin⁑𝑑+sin⁑𝑑+𝑑 .cπ‘œπ‘ β‘π‘‘ ) 𝒅𝒙/𝒅𝒕 = 𝒂 .𝒕.𝒄𝒐𝒔⁑𝒕 Finding π’…π’š/𝒅𝒙 π’…π’š/𝒅𝒙 = (π’…π’š/𝒅𝒕)/(𝒅𝒙/𝒅𝒕) 𝑑𝑦/𝑑π‘₯ = (π‘Ž" " .𝑑.sin⁑𝑑)/(π‘Ž" " .𝑑.cos⁑𝑑 ) π’…π’š/𝒅𝒙 = 𝒕𝒂𝒏⁑𝒕 Again Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯. 𝒅/𝒅𝒙 (π’…π’š/𝒅𝒙) = 𝒅(𝒕𝒂𝒏⁑𝒕)/𝒅𝒙 (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 𝑑(tan⁑𝑑)/𝑑π‘₯ (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 𝑑(tan⁑𝑑)/𝑑π‘₯ . 𝑑𝑑/𝑑𝑑 (𝑑^2 𝑦)/(𝑑π‘₯^2 ) =sec^2⁑𝑑 . 𝑑𝑑/𝑑π‘₯ (𝑑^2 𝑦)/(𝑑π‘₯^2 ) =sec^2⁑𝑑 Γ· 𝒅𝒙/𝒅𝒕 (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = sec^2⁑𝑑 Γ· 𝒂.𝒕.𝒄𝒐𝒔𝒕 (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = (sec^2⁑𝑑 )/(π‘Ž" " . 𝑑.cos⁑𝑑 ) "We have calculated" 𝒅𝒙/𝒅𝒕 " = " π‘Ž" ".𝑑.π‘π‘œπ‘ β‘π‘‘ (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = (sec^2⁑𝑑 )/(π‘Ž" " . 𝑑 Γ— 1/sec⁑𝑑 ) (𝒅^𝟐 π’š)/(𝒅𝒙^𝟐 ) = (〖𝒔𝒆𝒄〗^πŸ‘β‘π’• )/(𝒂" " . 𝒕) 2

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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