Examples
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Example 4 Important
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Example 6
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Example 9
Example 10
Example 11 Important
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Example 13 Important
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Example 15 Important
Example 16
Example 17 Important
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Example 19
Example 20 Important
Example 21
Example 22
Example 23
Example 24 Important
Example 25
Example 26 (i)
Example 26 (ii) Important
Example 26 (iii) Important
Example 26 (iv)
Example 27 Important
Example 28
Example 29 Important
Example 30 Important
Example 31
Example 32
Example 33 Important
Example 34 Important
Example 35
Example 36 Important
Example 37
Example 38 Important
Example 39 (i)
Example 39 (ii) Important
Example 40 (i)
Example 40 (ii) Important You are here
Example 40 (iii) Important
Example 41
Example 42 Important
Example 43
Question 1 Deleted for CBSE Board 2024 Exams
Question 2 Important Deleted for CBSE Board 2024 Exams
Question 3 Deleted for CBSE Board 2024 Exams
Question 4 Important Deleted for CBSE Board 2024 Exams
Question 5 Deleted for CBSE Board 2024 Exams
Question 6 Important Deleted for CBSE Board 2024 Exams
Last updated at June 2, 2023 by Teachoo
Example 40 (Method 1) Differentiate the following w.r.t. x. (ii) tan −1 (sin𝑥/( 1 +〖 cos〗〖𝑥 〗 )) Let 𝑓(𝑥) = tan −1 (𝒔𝒊𝒏𝒙/( 1 +〖 𝒄𝒐𝒔〗〖𝒙 〗 )) 𝑓(𝑥) = tan −1 ((𝟐 〖𝐬𝐢𝐧 〗〖𝒙/𝟐〗 〖 𝐜𝐨𝐬 〗〖𝒙/𝟐〗)/( 1+ (𝟐 〖𝐜𝐨𝐬〗^𝟐〖 𝒙/𝟐〗 − 𝟏))) = tan −1 ((2 〖sin 〗〖𝑥/2〗 〖 cos 〗〖𝑥/2 〗)/( 2 cos^2〖 𝑥/2〗 )) = tan −1 ((2 〖sin 〗〖𝑥/2〗)/( 2 cos〖𝑥/2〗 )) We know that sin 2θ = 2 sin θ cos θ Replacing θ by 𝜃/2 sin θ = 2 𝒔𝒊𝒏〖𝜽/𝟐〗 𝒄𝒐𝒔〖𝜽/𝟐〗 and cos 2θ = 2cos2 θ – 1 Replacing θ by 𝜃/2 cos θ = 2cos2 𝜽/𝟐 – 1 = tan −1 (〖sin 〗〖𝑥/2〗/( cos〖 𝑥/2〗 )) = tan −1 (〖tan 〗〖𝑥/2〗 ) = 𝒙/𝟐 𝒇(𝒙) = 𝒙/𝟐 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑓’(𝑥) = 1/2 (𝑑(𝑥))/𝑑𝑥 𝒇’(𝒙) = 𝟏/𝟐 (As tan^(−1)〖(tan𝜃)〗 =𝜃) Example 40 (Method 2) Differentiate the following w.r.t. x. (ii) tan −1 (sin𝑥/( 1 +〖 cos〗〖𝑥 〗 )) Let 𝑓(𝑥) = tan −1 (𝒔𝒊𝒏𝒙/( 1 +〖 𝒄𝒐𝒔〗〖𝒙 〗 )) Differentiating w.r.t x 𝑓^′ (𝑥) = 1/(1 + (𝒔𝒊𝒏𝒙/( 1 +〖 𝒄𝒐𝒔〗〖𝒙 〗 ))^2 ) (𝒔𝒊𝒏𝒙/( 1 +〖 𝒄𝒐𝒔〗〖𝒙 〗 ))^′ = 1/(((1 + cos𝑥 )^2 + sin^2𝑥)/(1 + cos𝑥 )^2 ) (((𝒔𝒊𝒏𝒙 )^′ (𝟏 + 𝒄𝒐𝒔 𝒙)−(𝟏 + 𝒄𝒐𝒔 𝒙)^′ 𝒔𝒊𝒏 𝒙)/( (1 +〖 𝒄𝒐𝒔〗𝒙 )^𝟐 )) = (1 + cos𝑥 )^2/((1 + cos𝑥 )^2 + sin^2𝑥 ) ((𝒄𝒐𝒔 𝒙(𝟏 + 𝒄𝒐𝒔 𝒙). − (− 𝒔𝒊𝒏 𝒙)𝒔𝒊𝒏 𝒙)/( (1 +〖 𝒄𝒐𝒔〗𝒙 )^𝟐 ))= (𝑐𝑜𝑠 𝑥 + 𝒄𝒐𝒔^𝟐 𝒙 + 𝒔𝒊𝒏^𝟐 𝒙)/(1 + 〖𝐜𝐨𝐬〗^𝟐𝒙 + 2 cos𝑥 + 〖𝒔𝒊𝒏〗^𝟐𝒙 ) = (𝒄𝒐𝒔 𝒙 +𝟏)/(1 + 1 + 2 cos𝑥 ) = (𝒄𝒐𝒔 𝒙 + 𝟏)/(2 + 2 cos𝑥 ) = (1 + 𝒄𝒐𝒔 𝒙)/(2(1 + cos𝑥) ) = 1/2
