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Example 2 (i)

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Example 2 (iv)

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Example 3 (i) Important

Example 3 (ii) Important

Example 4 (i)

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Example 5

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Example 8

Example 9

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Example 14

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Example 16

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Example 18

Example 19 (i) Important

Example 19 (ii)

Example 20 (i)

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Example 21 (i)

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Example 22 (i)

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Last updated at March 22, 2023 by Teachoo

Example 21 Compute derivative of (ii) g(x) = cot x g(x) = cot x = cos𝑥/sin𝑥 Let u = cos x & v = sin x ∴ g(x) = 𝑢/𝑣 So, g’(x) = (𝑢/𝑣)^′ Using quotient rule g’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = cos x u’ = – sin x & v = sin x v’ = cos x Now, f’(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑐𝑜𝑠〖𝑥=〖− 𝑠𝑖𝑛 〗𝑥 〗 ) (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑠𝑖𝑛〖𝑥=〖𝑐𝑜𝑠 〗𝑥 〗 ) = (−sin〖𝑥 (sin𝑥 ) −〖 cos〗〖𝑥 (cos〖𝑥)〗 〗 〗)/(〖𝑠𝑖𝑛〗^2 𝑥) = (−sin2〖𝑥 −〖 cos2〗〖𝑥 〗 〗)/(〖𝑠𝑖𝑛〗^2 𝑥) = (−(𝐬𝐢𝐧𝟐〖𝒙 + 〖 𝐜𝐨𝐬𝟐〗〖𝒙) 〗 〗)/(〖𝑠𝑖𝑛〗^2 𝑥) = (−𝟏)/(〖𝑠𝑖𝑛〗^2 𝑥) = –cosec2x Hence, f’(x) = –cosec2x (𝑈𝑠𝑖𝑛𝑔 𝑠𝑖𝑛2𝑥+𝑐𝑜𝑠2𝑥=1)