Ex 5.4, 1

Transcript

Ex 5.4, 1 Differentiate 𝑤.𝑟.𝑡. 𝑥 in , 𝑒﷮𝑥﷯﷮ sin﷮𝑥﷯﷯ Let 𝑦 = 𝑒﷮𝑥﷯﷮ sin﷮𝑥﷯﷯ Let 𝑢 = 𝑒﷮𝑥﷯ & 𝑣 = sin﷮𝑥﷯ ∴ 𝑦 = 𝑢﷮𝑣﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑 𝑦﷯﷮𝑑𝑥﷯ = 𝑑 𝑢﷮𝑣﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑢﷯﷮′﷯𝑣 − 𝑣 ﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑢﷯﷮′﷯𝑣 − 𝑣 ﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 𝑒﷮𝑥﷯﷯﷮𝑑𝑥﷯ . sin﷮𝑥﷯ − 𝑑 sin﷮𝑥﷯﷯ ﷮𝑑𝑥﷯ . 𝑒﷮𝑥﷯﷮ sin﷮𝑥﷯﷯﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮𝑥﷯ . sin﷮𝑥﷯ − cos﷮𝑥﷯ . 𝑒﷮𝑥﷯﷮ sin﷮𝑥﷯﷯﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮𝑥﷯ sin﷮𝑥﷯ − cos﷮𝑥﷯﷯﷮ sin﷮𝑥﷯﷯﷮2﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒆﷮𝒙﷯ 𝒔𝒊𝒏﷮𝒙﷯ − 𝒄𝒐𝒔﷮𝒙﷯﷯﷮ 𝒔𝒊𝒏﷮𝟐﷯﷮𝒙﷯﷯