CBSE Class 12 Sample Paper for 2025 Boards
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CBSE Class 12 Sample Paper for 2025 Boards
Last updated at Oct. 3, 2024 by Teachoo
This question is similar to Chapter 5 Class 12 Continuity and Differentiability - Ex 5.5
Please check the question here
Question 23 (B) Differentiate the following function with respect to x:(cos 𝑥)^𝑥; (where ├ 𝑥∈(0,𝜋/2)).Let 𝑦 = 〖(𝑐𝑜𝑠𝑥 ) 〗^𝑥 We use log differentiation Taking log both sides . log𝑦 = log〖 (𝑐𝑜𝑠𝑥 ) 〗^𝑥 𝒍𝒐𝒈𝒚 = 𝒙 . 𝒍𝒐𝒈 (𝒄𝒐𝒔𝒙 ) Differentiating both sides 𝑤.𝑟.𝑡.𝑥. (As 𝑙𝑜𝑔(𝑎^𝑏 )=𝑏 . 𝑙𝑜𝑔𝑎) 𝑑(log𝑦 )/𝑑𝑥 = (𝑑(𝑥 . log(cos𝑥 ) ) )/𝑑𝑥 𝑑(log𝑦 )/𝑑𝑦 . 𝑑𝑦/𝑑𝑥 = (𝑑(𝑥 . log(cos𝑥 ) ) )/𝑑𝑥 1/𝑦 (𝑑𝑦/𝑑𝑥) = (𝑑(𝑥 . log(cos𝑥 ) ) )/𝑑𝑥 1/𝑦 (𝑑𝑦/𝑑𝑥) = 𝑑𝑥/𝑑𝑥 log〖( 𝑐𝑜𝑠 𝑥)〗+𝑥 (𝑑(𝑙𝑜𝑔(𝑐𝑜𝑠𝑥 ) ) )/𝑑𝑥 1/𝑦 (𝑑𝑦/𝑑𝑥) = log〖(cos𝑥)〗+𝑥 × 1/cos𝑥 × (𝑐𝑜𝑠 𝑥)^′ 1/𝑦 (𝑑𝑦/𝑑𝑥) = log〖(cos𝑥)〗+𝑥/cos𝑥 × (−sin𝑥 ) Using product Rule As (𝑢𝑣)’ = 𝑢’𝑣 + 𝑣’𝑢 1/𝑦 (𝑑𝑦/𝑑𝑥) = log〖(cos𝑥)〗−𝑥 tan𝑥 𝑑𝑦/𝑑𝑥 = 𝑦[log(cos𝑥 )−𝑥 tan𝑥 ] Putting value of 𝑦 𝒅𝒚/𝒅𝒙 = (𝐜𝐨𝐬𝒙 )^𝒙 [𝒍𝒐𝒈(𝒄𝒐𝒔𝒙 )−𝒙 𝒕𝒂𝒏𝒙 ]
