Example 41 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Example 41 Find f ′(x) if 𝑓 (𝑥) = (sin⁡𝑥 )^(sin 𝑥)⁡ for all 0 < x < π.Let 〖𝑦=(sin⁡𝑥 )〗^(sin 𝑥) Taking log on both sides log 𝑦=log (〖sin⁡𝑥〗^sin⁡𝑥 ) 𝐥𝐨𝐠 𝒚=𝒔𝒊𝒏⁡𝒙 . 𝐥𝐨𝐠 (𝒔𝒊𝒏⁡𝒙 ) Differentiating both sides 𝑤.𝑟.𝑡. x 𝑑(log⁡𝑦)/𝑑𝑥 = 𝑑(sin⁡𝑥 . log (sin⁡𝑥 ))/𝑑𝑥 𝑑(log⁡𝑦 )/𝑑𝑥 (𝑑𝑦/𝑑𝑦) = 𝑑(sin⁡𝑥 . log (sin⁡𝑥 ))/𝑑𝑥 (As 𝑙𝑜𝑔⁡(𝑎^𝑏 )=𝑏 . 𝑙𝑜𝑔⁡𝑎) 𝑑(log⁡𝑦 )/𝑑𝑦 (𝑑𝑦/𝑑𝑥) = 𝑑(sin⁡𝑥 . log (sin⁡𝑥 ))/𝑑𝑥 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 𝑑(sin⁡𝑥 . log (sin⁡𝑥 ))/𝑑𝑥 1/𝑦 𝑑𝑦/𝑑𝑥 = 𝑑(sin⁡𝑥 )/𝑑𝑥 . 〖 log 〗⁡(sin⁡𝑥 ) + 𝑑(〖log 〗⁡(sin⁡𝑥 ) )/𝑑𝑥 . sin⁡𝑥 1/𝑦 𝑑𝑦/𝑑𝑥 = cos⁡𝑥 . 〖log 〗⁡(sin⁡𝑥 ) + 1/sin⁡𝑥 . 𝑑(sin⁡𝑥 )/𝑑𝑥 . sin⁡𝑥 1/𝑦 𝑑𝑦/𝑑𝑥 = 〖cos 〗⁡. 〖log 〗⁡(sin⁡𝑥 ) + 1/sin⁡𝑥 . cos⁡𝑥.sin 𝑥 1/𝑦 𝑑𝑦/𝑑𝑥 = cos⁡𝑥.log sin 𝑥+cos⁡𝑥 Using Product rule (uv)’ = u’v + v’u where u = sin x & v = log (sin x) 𝑑𝑦/𝑑𝑥 = 𝑦 (cos⁡𝑥.log sin 𝑥+cos⁡𝑥 ) Putting value of y = (𝑠𝑖𝑛⁡𝑥 )^𝑠𝑖𝑛⁡𝑥 𝑑𝑦/𝑑𝑥 = (sin⁡𝑥 )^sin⁡𝑥 (cos⁡𝑥.log sin 𝑥+cos⁡𝑥 ) 𝑑𝑦/𝑑𝑥 = (sin⁡𝑥 )^sin⁡𝑥 . cos⁡𝑥 (log sin 𝑥+1) 𝒅𝒚/𝒅𝒙 = 〖(𝟏+𝐥𝐨𝐠 (𝐬𝐢𝐧 𝒙)) (𝒔𝒊𝒏⁡𝒙 )〗^𝒔𝒊𝒏⁡𝒙 . 𝒄𝒐𝒔⁡𝒙