Example 45 - Differentiate (i) cos-1 (sin x) - Chapter 5 - Logarithmic Differentiation - Type 1

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Example 45 Differentiate the following 𝑤.𝑟.𝑡. 𝑥. (i) cos﷮−1﷯ (sin⁡𝑥) Let 𝑓(𝑥) = cos﷮−1﷯ (sin⁡𝑥) 𝑓(𝑥) = cos﷮−1﷯ cos ﷮ 𝜋﷮2﷯ −𝑥﷯﷯﷯ 𝒇(𝒙) = 𝝅﷮𝟐﷯ −𝒙 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑓’(𝑥) = 𝑑 𝜋﷮2﷯﷯﷮𝑑𝑥﷯ − 𝑑(𝑥)﷮𝑑𝑥﷯ 𝑓’(𝑥) = 0 − 1 𝒇’(𝒙) = − 1 Example 45 Differentiate the following w.r.t. x. (ii) tan −1 sin﷮𝑥﷯﷮ 1 + cos﷮𝑥 ﷯﷯﷯ Let 𝑓(𝑥) = tan −1 sin﷮𝑥﷯﷮ 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﷯﷯ ﷯﷯ = 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﷯ Example 45 Differentiate the following w.r.t. x. (iii) sin﷮−1﷯ 2﷮ 𝑥+1﷯ ﷮ 1 + 4 ﷮𝑥﷯﷯﷯ Let 𝑓 𝑥﷯ = sin﷮−1﷯ 2﷮ 𝑥+1﷯ ﷮ 1 + 4 ﷮𝑥﷯﷯﷯ 𝑓(𝑥) = sin﷮−1﷯ 2﷮ 𝑥﷯. 2﷮ 1 + 2﷮𝑥﷯﷯﷮2﷯﷯﷯ Let 𝟐﷮𝒙﷯ = tan θ 𝑓(𝑥) = sin﷮−1﷯ tan﷮𝜃 ﷯. 2﷮ 1 + tan﷮2﷯﷮𝜃﷯﷯﷯ = sin﷮−1﷯ 2 tan﷮𝜃 ﷯ ﷮ 1 + tan﷮2﷯﷮𝜃﷯﷯﷯ = sin﷮−1﷯ (sin 2𝜃) = 2𝜃 ∴ 𝑓(𝑥) = 2𝜃 As 2﷮𝑥﷯= tan⁡𝜃 tan﷮−1﷯ 2﷮𝑥﷯﷯=𝜃 ∴ 𝒇(𝒙) = 𝟐 𝒕𝒂𝒏﷮−𝟏﷯ 𝟐﷮𝒙﷯﷯﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 𝑓’(𝑥) = 2 𝑑 tan﷮−1﷯ 2﷮𝑥﷯﷯ ﷮𝑑𝑥﷯ 𝑓’(𝑥) = 2 . 1﷮1 + 2﷮𝑥﷯﷯﷮2﷯ ﷯ . 𝒅 𝟐﷮𝒙﷯﷯ ﷮𝒅𝒙﷯ 𝑓’(𝑥) = 2 ﷮1 + 2﷮𝑥﷯﷯﷮2﷯﷯ . 𝟐﷮𝒙﷯ . 𝒍𝒐𝒈⁡𝟐 𝑓’(𝑥) = 2. 2﷮𝑥﷯. log﷮2﷯﷮1 + 2﷮𝑥﷯﷯﷮2﷯﷯ 𝑓’(𝑥) = 2﷮𝑥 + 1﷯. log﷮2﷯﷮1 + 2﷮𝑥﷯﷯﷮2﷯﷯ 𝑓’(𝑥) = 2﷮𝑥 + 1﷯. log﷮2﷯﷮1 + 2﷮2﷯﷯﷮𝑥﷯﷯ 𝒇’(𝒙) = 𝟐﷮𝒙 + 𝟏﷯. 𝒍𝒐𝒈﷮𝟐﷯﷮𝟏 + 𝟒﷮𝒙﷯﷯

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