Example 43 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Example 43 Differentiate 〖𝑠𝑖𝑛〗^2 𝑥 𝑤.𝑟.𝑡. 𝑒^(cos⁡𝑥 ". " )Let 𝑢 = 〖𝑠𝑖𝑛〗^2 𝑥 & 𝑣 =𝑒^(cos⁡𝑥 ) We need to differentiate 𝑢 𝑤.𝑟.𝑡. 𝑣 . i.e., 𝑑𝑢/𝑑𝑣 Here, 𝒅𝒖/𝒅𝒗 = (𝒅𝒖/𝒅𝒙)/(𝒅𝒗/𝒅𝒙) Calculating 𝒅𝒖/𝒅𝒙 𝑢 = 〖𝑠𝑖𝑛〗^2 𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑢/𝑑𝑥 = 𝑑(〖𝑠𝑖𝑛〗^2 𝑥)/𝑑𝑥 𝑑𝑢/𝑑𝑥 = 2 sin⁡𝑥 . 𝑑(sin⁡𝑥 )/𝑑𝑥 𝑑𝑢/𝑑𝑥 = 𝟐 𝒔𝒊𝒏⁡𝒙 . 𝐜𝐨𝐬⁡𝒙 Calculating 𝒅𝒗/𝒅𝒙 𝑣 =𝑒^(cos⁡𝑥 ) Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑣/𝑑𝑥 = 𝑑(𝑒^(cos⁡𝑥 ) )/𝑑𝑥 𝑑𝑣/𝑑𝑥 = 𝑒^(cos⁡𝑥 ) . 𝑑(cos⁡𝑥 )/𝑑𝑥 𝑑𝑣/𝑑𝑥 = 𝑒^(cos⁡𝑥 ) . (−sin⁡𝑥 ) 𝑑𝑣/𝑑𝑥 = −𝒔𝒊𝒏⁡𝒙. 𝒆^(𝒄𝒐𝒔⁡𝒙 )Therefore 𝑑𝑢/𝑑𝑣 = (𝑑𝑢/𝑑𝑥)/(𝑑𝑣/𝑑𝑥) = (2 sin⁡𝑥" ." cos⁡𝑥)/(−sin⁡𝑥 . 𝑒^(cos⁡𝑥 ) ) = (−𝟐"." 𝒄𝒐𝒔⁡𝒙)/𝒆^(𝒄𝒐𝒔⁡𝒙 )