Check sibling questions

    Slide6.JPG

Slide7.JPG
Slide8.JPG Slide9.JPG Slide10.JPG

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


Transcript

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 log⁡𝑢=log⁡〖𝑥^𝑥 〗 log⁡𝑢=𝑥 .log⁡𝑥 (𝑎^𝑎 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑(log⁡𝑢 )/𝑑𝑥 = 𝑑(𝑥 .〖 log〗⁡𝑥 )/𝑑𝑥 𝑑(log⁡𝑢 )/𝑑𝑥 . 𝑑𝑢/𝑑𝑢 = 𝑑(𝑥 .〖 log〗⁡𝑥 )/𝑑𝑥 𝑑(log⁡𝑢 )/𝑑𝑢 . 𝑑𝑢/𝑑𝑥 = 𝑑𝑥/𝑑𝑥 . log⁡𝑥 + (𝑑(〖 log〗⁡𝑥))/𝑑𝑥. 𝑥 1/𝑢 . 𝑑𝑢/𝑑𝑥 = log⁡𝑥 + 1/𝑥 . 𝑥 1/𝑢 . 𝑑𝑢/𝑑𝑥 = log⁡𝑥 + 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 .log⁡𝑎 Therefore, 𝑑𝑦/𝑑𝑥 = 𝑑𝑢/𝑑𝑥 + 𝑑𝑣/𝑑𝑥 + 𝑑𝑤/𝑑𝑥 = 𝒙^𝒙 (𝟏+ 𝒍𝒐𝒈 𝒙) + 𝒂𝒙^(𝒂 −𝟏) + 𝒂^𝒙 .𝒍𝒐𝒈⁡𝒂

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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.