Example 32 - Differentiate xsin x - Chapter 5 Class 12 - Examples

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Example 32 Differentiate 𝑥﷮ sin﷮𝑥﷯﷯ , 𝑥 > 0 𝑤.𝑟.𝑡. 𝑥. Let y = 𝑥﷮ sin﷮𝑥﷯﷯ Taking log both sides log⁡𝑦 = log 𝑥﷮ sin﷮𝑥﷯﷯ log⁡𝑦 = sin﷮𝑥﷯ . log 𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑( log﷮𝑦)﷯﷮𝑑𝑥﷯ = 𝑑﷮𝑑𝑥﷯ sin﷮𝑥 log﷮𝑥﷯﷯﷯ 𝑑( log﷮𝑦)﷯﷮𝑑𝑥﷯ = 𝑑( sin﷮𝑥﷯)﷮𝑑𝑥﷯.log 𝑥+sin 𝑥 . 𝑑( log﷮𝑥﷯)﷮𝑑𝑥﷯ 𝑑( log﷮𝑦)﷯﷮𝑑𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡𝑥 log⁡𝑥 + sin⁡𝑥 1﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ 1﷮𝑦﷯ = cos 𝑥⁡log⁡𝑥 + sin⁡𝑥 1﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦 cos x﷮ log﷮𝑥+ 1﷮𝑥﷯﷯ sin﷮𝑥﷯﷯﷯ Putting back 𝑦 = 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ cos﷮ log﷮𝑥+ 1﷮𝑥﷯﷯ sin﷮𝑥﷯﷯﷯ = 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ cos﷮ log﷮𝑥﷯﷯ + 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ 1﷮𝑥﷯ 𝑠𝑖𝑛﷮𝑥﷯ = 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ cos﷮ log﷮𝑥﷯﷯ + 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ 𝑥﷮−1﷯ sin﷮𝑥﷯ = 𝑥﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ cos﷮ log﷮𝑥﷯﷯ + 𝑥﷮ 𝑠𝑖𝑛﷮𝑥−1﷯﷯ sin﷮𝑥﷯ = 𝒙﷮ 𝒔𝒊𝒏﷮𝒙﷯−𝟏﷯. 𝒔𝒊𝒏﷮𝒙﷯ + 𝒙﷮ 𝒔𝒊𝒏﷮𝒙﷯﷯. 𝐜𝐨𝐬 𝒙 𝐥𝐨𝐠 𝒙

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