Example 29  - Differentiate w.r.t. x: (i) e-x (ii) sin log x - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 29 Differentiate the following w.r.t. x: (i) 𝑒﷮–𝑥﷯ Let 𝑦 = 𝑒﷮–𝑥﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑 𝑦﷯﷮𝑑𝑥﷯ = 𝑑 𝑒﷮−𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑 𝑦﷯﷮𝑑𝑥﷯ = 𝑒﷮−𝑥﷯ . 𝑑 −𝑥﷯﷮𝑑𝑥﷯ 𝑑 𝑦﷯﷮𝑑𝑥﷯ = 𝑒﷮−𝑥﷯ (−1) . 𝒅 𝒚﷯﷮𝒅𝒙﷯ = −𝒆﷮−𝒙﷯ Example 29 Differentiate the following w.r.t. x: (ii) sin⁡(log⁡𝑥), 𝑥 > 0 Let 𝑦 =sin⁡(log⁡𝑥) Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 sin⁡( log﷮𝑥﷯)﷯ ﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡(log⁡𝑥) . 𝑑﷮ log﷮𝑥﷯﷯﷯ ﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡(log⁡𝑥) . 1﷮𝑥﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒄𝒐𝒔⁡(𝒍𝒐𝒈⁡𝒙) ﷮𝒙﷯ Example 29 Differentiate the following w.r.t. x: (iii) 𝑐𝑜𝑠﷮−1﷯(ex) Let 𝑦 = 𝑐𝑜𝑠﷮−1﷯(ex) Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 cos﷮−1﷯﷮ 𝑒﷮𝑥﷯﷯﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = −1﷮ ﷮1 − 𝑒﷮𝑥﷯﷯﷮2﷯ ﷯﷯ . 𝑑 𝑒﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = −1﷮ ﷮1 − 𝑒﷮𝑥﷯﷯﷮2﷯ ﷯﷯ . 𝑒﷮𝑥﷯ 𝒅𝒚﷮𝒅𝒙﷯ = − 𝒆﷮𝒙﷯﷮ ﷮𝟏 − 𝒆﷮𝟐𝒙﷯﷯﷯ Example 29 Differentiate the following w.r.t. x: (iv) 𝑒𝑐𝑜𝑠 𝑥 Let 𝑦 = 𝑒﷮ cos﷮𝑥﷯﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 𝑒﷮ cos﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮ cos﷮𝑥﷯﷯ . 𝑑 cos﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮ cos﷮𝑥﷯﷯ . − sin﷮𝑥﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = − 𝒔𝒊𝒏﷮𝒙﷯. 𝒆﷮ 𝒄𝒐𝒔﷮𝒙﷯﷯

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