Check sibling questions

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


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Misc 11 Differentiate w.r.t. x the function, 𝑥^(𝑥^2− 3)+(𝑥−3)𝑥^2, for 𝑥 > 3 Let 𝑦=𝑥^(𝑥^2− 3)+(𝑥−3)^(𝑥^2 ) And let 𝑢=𝑥^(𝑥^2− 3) , 𝑣 =(𝑥−3)^(𝑥^2 ) Now, 𝒚 = 𝒖+𝒗 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦/𝑑𝑥 = (𝑑 (𝑢 + 𝑣))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑑𝑢/𝑑𝑥 + 𝑑𝑣/𝑑𝑥 Calculating 𝒅𝒖/𝒅𝒙 𝑢 = 𝑥^(𝑥^2− 3) Taking log on both sides log 𝑢=log⁡〖𝑥^(𝑥^2− 3) 〗 log 𝑢=〖(𝑥〗^2− 3). log⁡𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑(log⁡𝑢 )/𝑑𝑥 = 𝑑(〖(𝑥〗^2− 3) log⁡𝑥 )/𝑑𝑥 𝑑(log⁡𝑢 )/𝑑𝑥 . 𝑑𝑢/𝑑𝑢 = 𝑑(〖(𝑥〗^2 − 3) log⁡𝑥 )/𝑑𝑥 " " 𝑑(log⁡𝑢 )/𝑑𝑢 . 𝑑𝑢/𝑑𝑥 = 𝑑(〖(𝑥〗^2− 3) log⁡𝑥 )/𝑑𝑥 " " 1/𝑢 . 𝑑𝑢/𝑑𝑥 = 𝑑(〖(𝑥〗^2− 3) log⁡𝑥 )/𝑑𝑥 1/𝑢 . 𝑑𝑢/𝑑𝑥 = (𝑑〖(𝑥〗^2− 3) )/𝑑𝑥 . 〖 log〗⁡𝑥 + 𝑑(log⁡𝑥 )/𝑑𝑥 . 〖(𝑥〗^2− 3) 1/𝑢 . 𝑑𝑢/𝑑𝑥 = (2𝑥 −0) 〖 log〗⁡𝑥 + 1/𝑥 × 〖(𝑥〗^2− 3) 1/𝑢 . 𝑑𝑢/𝑑𝑥 = 2𝑥 . log⁡𝑥 + (𝑥^2− 3)/𝑥 𝑑𝑢/𝑑𝑥 = u (2𝑥 "." log⁡𝑥 "+ " (𝑥^2− 3)/𝑥) 𝒅𝒖/𝒅𝒙 = 𝒙^(𝒙^𝟐− 𝟑) (𝟐𝒙 "." 𝒍𝒐𝒈⁡𝒙 "+ " (𝒙^𝟐− 𝟑)/𝒙) Calculating 𝒅𝒗/𝒅𝒙 𝑣 = (𝑥−3)𝑥^2 Taking log on both sides log 𝑣=log⁡〖(𝑥−3)^(𝑥^2 ) 〗 log 𝑣=〖𝑥^2 . log〗⁡〖 (𝑥−3)〗 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑(log⁡𝑣 )/𝑑𝑥 = (𝑑(〖𝑥^2. log〗⁡〖 (𝑥 − 3)〗 ) )/𝑑𝑥 𝑑(log⁡𝑣 )/𝑑𝑥 . 𝑑𝑣/𝑑𝑣 = (𝑑(〖𝑥^2. log〗⁡〖 (𝑥−3)〗 ) )/𝑑𝑥 𝑑(log⁡𝑣 )/𝑑𝑣 . 𝑑𝑣/𝑑𝑥 = (𝑑(〖𝑥^2. log〗⁡〖 (𝑥−3)〗 ) )/𝑑𝑥 1/𝑣 . 𝑑𝑣/𝑑𝑥 = (𝑑(〖𝑥^2. log〗⁡〖 (𝑥−3)〗 ) )/𝑑𝑥 1/𝑣 . 𝑑𝑣/𝑑𝑥 = 𝑑(𝑥^2 )/𝑑𝑥 . log (𝑥−3) + 𝑑(log" " (𝑥 − 3))/𝑑𝑥 . 𝑥^2 1/𝑣 . 𝑑𝑣/𝑑𝑥 = 2𝑥 . log (𝑥−3) + 1/((𝑥 − 3) ). (𝑑(𝑥 − 3)" " )/𝑑𝑥 . 𝑥^2 1/𝑣 . 𝑑𝑣/𝑑𝑥 = 2𝑥 . log (𝑥−3) + 1/((𝑥 − 3) ) . 𝑥^2 1/𝑣 . 𝑑𝑣/𝑑𝑥 = 2𝑥. log (𝑥−3) + 𝑥^2/(𝑥 −3) 𝑑𝑣/𝑑𝑥 = 𝑣 (2𝑥". " log" " (𝑥−3)" + " 𝑥^2/(𝑥 −3)) 𝒅𝒗/𝒅𝒙 = (𝒙−𝟑)𝒙^𝟐 (𝟐𝒙". " 𝐥𝐨𝐠" " (𝒙−𝟑)" + " 𝒙^𝟐/(𝒙 −𝟑)) Now, 𝑑𝑦/𝑑𝑥 = 𝑑𝑢/𝑑𝑥 + 𝑑𝑣/𝑑𝑥 = 𝒙^(𝒙^𝟐− 𝟑) ((𝒙^𝟐− 𝟑)/𝒙+𝟐𝒙 𝐥𝐨𝐠⁡𝒙 ) + (𝒙−𝟑)𝒙^𝟐 (𝒙^𝟐/(𝒙 −𝟑)+𝟐𝒙 .𝐥𝐨𝐠⁡(𝒙 −𝟑) )

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.