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Miscellaneous
Misc 2
Misc 3
Misc 4
Misc 5 Important
Misc 6 Important
Misc 7 Important
Misc 8
Misc 9 Important
Misc 10 You are here
Misc 11 Important
Misc 12
Misc 13 Important
Misc 14 Important
Misc 15 Important
Misc 16 Important
Misc 17 Important
Misc 18
Misc 19
Misc 20
Misc 21
Misc 22 Important
Question 1 Important Deleted for CBSE Board 2024 Exams
Last updated at June 5, 2023 by Teachoo
Misc 10 Differentiate w.r.t. x the function, 𝑥𝑥 + 𝑥𝑎 + 𝑎^𝑥+ 𝑎𝑎, for some fixed 𝑎 >0 and 𝑥> 0Let 𝑦= 𝑥𝑥 + 𝑥𝑎 + 𝑎^𝑥+ 𝑎𝑎 And let u=𝑥𝑥 , 𝑣=𝑥𝑎 , 𝑤=𝑎^𝑥 Now, 𝒚=𝒖+𝒗+𝒘+𝒂𝒂 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦/𝑑𝑥 = 𝑑(𝑢 + 𝑣 + 𝑤 + 𝑎𝑎)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑑(𝑢)/𝑑𝑥 +𝑑(𝑣)/𝑑𝑥+𝑑(𝑤)/𝑑𝑥 + 𝑑(𝑎𝑎)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑑𝑢/𝑑𝑥 +𝑑𝑣/𝑑𝑥+𝑑𝑤/𝑑𝑥 + 0 𝑑𝑦/𝑑𝑥 = 𝑑𝑢/𝑑𝑥 + 𝑑𝑣/𝑑𝑥 + 𝑑𝑤/𝑑𝑥 Calculating 𝒅𝒖/𝒅𝒙 𝑢 =𝑥^𝑥 Taking log on both sides log𝑢=log〖𝑥^𝑥 〗 log𝑢=𝑥 .log𝑥 (𝑎^𝑎 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑(log𝑢 )/𝑑𝑥 = 𝑑(𝑥 .〖 log〗𝑥 )/𝑑𝑥 𝑑(log𝑢 )/𝑑𝑥 . 𝑑𝑢/𝑑𝑢 = 𝑑(𝑥 .〖 log〗𝑥 )/𝑑𝑥 𝑑(log𝑢 )/𝑑𝑢 . 𝑑𝑢/𝑑𝑥 = 𝑑𝑥/𝑑𝑥 . log𝑥 + (𝑑(〖 log〗𝑥))/𝑑𝑥. 𝑥 1/𝑢 . 𝑑𝑢/𝑑𝑥 = log𝑥 + 1/𝑥 . 𝑥 1/𝑢 . 𝑑𝑢/𝑑𝑥 = log𝑥 + 1 𝑑𝑢/𝑑𝑥 = u (1+ log 𝑥) 𝒅𝒖/𝒅𝒙 = 𝒙^𝒙 (𝟏+ 𝒍𝒐𝒈 𝒙) Calculating 𝒅𝒗/𝒅𝒙 𝑣=𝑥^𝑎 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑣/𝑑𝑥= 𝑑(𝑥^𝑎 )/𝑑𝑥 𝒅𝒗/𝒅𝒙= 𝒂𝒙^(𝒂 −𝟏) Calculating 𝒅𝒘/𝒅𝒙 𝑤=𝑎^𝑥 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑤/𝑑𝑥 = 𝑑(𝑎^𝑥 )/𝑑𝑥 𝑑𝑤/𝑑𝑥 = 𝑎^𝑥 .log𝑎 Therefore, 𝑑𝑦/𝑑𝑥 = 𝑑𝑢/𝑑𝑥 + 𝑑𝑣/𝑑𝑥 + 𝑑𝑤/𝑑𝑥 = 𝒙^𝒙 (𝟏+ 𝒍𝒐𝒈 𝒙) + 𝒂𝒙^(𝒂 −𝟏) + 𝒂^𝒙 .𝒍𝒐𝒈𝒂