Ex 5.4, 6

Transcript

Ex 5.4, 6 Differentiate 𝑤.𝑟.𝑡. 𝑥 in , 𝑒﷮𝑥﷯+ 𝑒﷮ 𝑥﷮2﷯﷯ + 𝑒﷮ 𝑥﷮3﷯﷯+ 𝑒﷮ 𝑥﷮4﷯﷯ + 𝑒﷮ 𝑥﷮5﷯﷯ Let 𝑦 = 𝑒﷮𝑥﷯+ 𝑒﷮ 𝑥﷮2﷯﷯ +... + 𝑒﷮ 𝑥﷮5﷯﷯ y = 𝑒﷮𝑥﷯+ 𝑒﷮ 𝑥﷮2﷯﷯ + 𝑒﷮ 𝑥﷮3﷯﷯+ 𝑒﷮ 𝑥﷮4﷯﷯ + 𝑒﷮ 𝑥﷮5﷯﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑 𝑦﷯﷮𝑑𝑥﷯ = 𝑑 𝑒﷮𝑥﷯+ 𝑒﷮ 𝑥﷮2﷯﷯ + 𝑒﷮ 𝑥﷮3﷯﷯+ 𝑒﷮ 𝑥﷮4﷯﷯ + 𝑒﷮ 𝑥﷮5﷯﷯﷯﷮𝑑𝑥﷯ = 𝑑 𝑒﷮𝑥﷯﷯ ﷮𝑑𝑥﷯ + 𝑑 𝑒﷮ 𝑥﷮2﷯﷯﷯﷮𝑑𝑥﷯ + 𝑑 𝑒﷮ 𝑥﷮3﷯﷯﷯﷮𝑑𝑥﷯ + 𝑑 𝑒﷮ 𝑥﷮4﷯﷯﷯﷮𝑑𝑥﷯ + 𝑑 𝑒﷮ 𝑥﷮5﷯﷯﷯﷮𝑑𝑥﷯ = 𝑒﷮𝑥﷯ + 𝑒﷮ 𝑥﷮2﷯﷯. 𝑑 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ + 𝑒﷮ 𝑥﷮3﷯﷯. 𝑑 𝑥﷮3﷯﷯﷮𝑑𝑥﷯ + 𝑒﷮ 𝑥﷮4﷯﷯. 𝑑 𝑥﷮4﷯﷯﷮𝑑𝑥﷯ + 𝑒﷮ 𝑥﷮5﷯﷯. 𝑑 𝑥﷮5﷯﷯﷮𝑑𝑥﷯ = 𝒆﷮𝒙﷯ + 2x 𝒆﷮ 𝒙﷮𝟐﷯﷯ + 𝟑 𝒙﷮𝟐﷯ 𝒆﷮ 𝒙﷮𝟑﷯﷯ + 𝟒 𝒙﷮𝟑﷯ 𝒆﷮ 𝒙﷮𝟒﷯﷯ + 𝟓𝒙﷮𝟒 ﷯ 𝒆﷮ 𝒙﷮𝟓﷯﷯