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Ex 5.5
Ex 5.5, 2
Ex 5.5, 3 Important
Ex 5.5, 4
Ex 5.5, 5
Ex 5.5,6 Important
Ex 5.5, 7 Important
Ex 5.5, 8
Ex 5.5, 9 Important
Ex 5.5, 10 Important
Ex 5.5, 11 Important
Ex 5.5, 12
Ex 5.5, 13 You are here
Ex 5.5, 14 Important
Ex 5.5, 15
Ex 5.5, 16 Important
Ex 5.5, 17 Important
Ex 5.5, 18
Last updated at June 12, 2023 by Teachoo
Ex 5.5, 13 Find 𝑑𝑦/𝑑𝑥 of the functions in, 𝑦^𝑥 = 𝑥^𝑦 Given, 𝑦^𝑥 = 𝑥^𝑦 Taking log both sides log (𝑦^𝑥 ) = log (𝑥^𝑦 ) 𝑥 . log 𝑦=𝑦.log𝑥 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. (𝑑(𝑥 . log 𝑦))/𝑑𝑥 = 𝑑(𝑦.〖 log〗𝑥 )/𝑑𝑥 (As 𝑙𝑜𝑔(𝑎^𝑏 )=𝑏 . 𝑙𝑜𝑔𝑎) Using product Rule As (𝑢𝑣)’ = 𝑢’𝑣 + 𝑣’𝑢 𝑑(𝑥)/𝑑𝑥 . log 𝑦+ 𝑑(log𝑦 )/𝑑𝑥 . 𝑥 =" " 𝑑(𝑦)/𝑑𝑥 " ". log 𝑥 + 𝑑(log𝑥 )/𝑑𝑥 . 𝑦 log 𝑦+𝑥 . 𝑑(log𝑦 )/𝑑𝑥 . 𝑥 = 𝑑𝑦/𝑑𝑥 log 𝑥 + 1/𝑥 . 𝑦 log 𝑦+𝑥 . 𝑑(log𝑦 )/𝑑𝑥 . 𝑑𝑦/𝑑𝑦 = 𝑑𝑦/𝑑𝑥 . log 𝑥 + 𝑦/𝑥 log 𝑦+𝑥 . 𝑑(log𝑦 )/𝑑𝑦 . 𝑑𝑦/𝑑𝑥 = 𝑑𝑦/𝑑𝑥 . log 𝑥 + 𝑦/𝑥 log 𝑦+𝑥 . 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 𝑑𝑦/𝑑𝑥 . log 𝑥 + 𝑦/𝑥 log 𝑦+ 𝑥/𝑦 . 𝑑𝑦/𝑑𝑥 = 𝑑𝑦/𝑑𝑥 . log 𝑥 + 𝑦/𝑥 𝑥/𝑦 . 𝑑𝑦/𝑑𝑥 − 𝑑𝑦/𝑑𝑥 . log 𝑥 = 𝑦/𝑥 − log 𝑦 𝑑𝑦/𝑑𝑥 (𝑥/𝑦 − log 𝑥) = 𝑦/𝑥 − log 𝑦 𝑑𝑦/𝑑𝑥 ((𝑥 − 𝑦 log𝑥)/𝑦) = (𝑦 − 𝑥 log𝑦)/𝑥 𝑑𝑦/𝑑𝑥 = (𝑦 − 𝑥 log𝑦)/𝑥 . 𝑦/(𝑥 − 𝑦 log𝑥 ) 𝒅𝒚/𝒅𝒙 = 𝒚(𝒚 − 𝒙 𝒍𝒐𝒈𝒚 )/𝒙(𝒙 − 𝒚 𝒍𝒐𝒈𝒙 )