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Misc  11 - Differentiate the function - x^(x^2-3) + (x-3)^(x^2)

Misc  11 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Misc  11 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Misc  11 - Chapter 5 Class 12 Continuity and Differentiability - Part 4 Misc  11 - Chapter 5 Class 12 Continuity and Differentiability - Part 5 Misc  11 - Chapter 5 Class 12 Continuity and Differentiability - Part 6

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Misc 11 Differentiate w.r.t. x the function, 𝑥^(𝑥^2− 3)+(𝑥−3)𝑥^2, for 𝑥 > 3Let 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 " " (As 𝑙𝑜𝑔⁡(𝑎^𝑏) = 𝑏 𝑙𝑜𝑔⁡𝑎) 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) Using Product rule As (𝑢𝑣)’ = 𝑢’𝑣 + 𝑣’𝑢 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 (As log⁡(𝑎^𝑏) = 𝑏 log⁡𝑎) 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)Using product rule (𝑢𝑣)’ = 𝑢’𝑣 + 𝑣’𝑢 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.