Ex 5.5, 5

Transcript

Ex 5.5, 5 Differentiate the functions in, 𝑥 + 3﷯﷮2﷯ . 𝑥 + 4﷯﷮3﷯ . 𝑥 + 5﷯﷮4﷯ Let 𝑦= 𝑥 + 3﷯﷮2﷯ . 𝑥 + 4﷯﷮3﷯ . 𝑥 + 5﷯﷮4﷯ Taking log both sides log⁡𝑦 = log 𝑥 + 3﷯﷮2﷯ . 𝑥 + 4﷯﷮3﷯ . 𝑥 + 5﷯﷮4﷯﷯ log⁡𝑦 = log 𝑥 + 3﷯﷮2﷯ + log 𝑥 + 4﷯﷮3﷯ + log 𝑥 + 5﷯﷮4﷯ log⁡𝑦 = 2 log 𝑥 + 3﷯ + 3 log 𝑥 + 4﷯ + 4 log 𝑥 + 5﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ = 𝑑 2 log 𝑥 + 3﷯ + 3 log 𝑥 + 4﷯ + 4 log 𝑥 + 5﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑦﷯﷯ = 𝑑 2 log 𝑥 + 3﷯﷯﷮𝑑𝑥﷯ + 𝑑 3 log 𝑥 + 4﷯﷯﷮𝑑𝑥﷯ + 𝑑 4 log 𝑥 + 5﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑦﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 2 𝑑 log 𝑥 + 3﷯﷯﷮𝑑𝑥﷯ + 3 𝑑 log 𝑥 + 4﷯﷯﷮𝑑𝑥﷯ + 4 𝑑 log 𝑥 + 5﷯﷯﷮𝑑𝑥﷯ 1﷮𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯ = 2. 1﷮ 𝑥 + 3﷯﷯ . 𝑑 𝑥 + 3﷯﷮𝑑𝑥﷯ + 3. 1﷮ 𝑥 + 4﷯﷯ . 𝑑 𝑥 + 4﷯﷮𝑑𝑥﷯ + 4. 1﷮ 𝑥 + 5﷯﷯ . 𝑑 𝑥 + 5﷯﷮𝑑𝑥﷯ 1﷮𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯ = 2﷮𝑥 + 3﷯ 𝑑𝑥﷮𝑑𝑥﷯+ 𝑑 3﷯﷮𝑑𝑥﷯﷯ + 3﷮𝑥 + 4﷯ 𝑑𝑥﷮𝑑𝑥﷯+ 𝑑 4﷯﷮𝑑𝑥﷯﷯ + 4﷮𝑥 +5﷯ 𝑑𝑥﷮𝑑𝑥﷯+ 𝑑 5﷯﷮𝑑𝑥﷯﷯ 1﷮𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯ = 2﷮𝑥 + 3﷯ 1+0﷯ + 3﷮𝑥 + 4﷯ 1+0﷯ + 4﷮𝑥 + 5﷯ 1+0﷯ 1﷮𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯ = 2﷮𝑥 + 3﷯ + 3﷮𝑥 + 4﷯ + 4﷮𝑥 + 5﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦 2﷮𝑥 + 3﷯ + 3﷮𝑥 + 4﷯ + 4﷮𝑥 + 5﷯﷯ Putting value of 𝑦 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 3﷯﷮2﷯ . 𝑥 + 4﷯﷮3﷯ . 𝑥 + 5﷯﷮4 ﷯ 2﷮ 𝑥 + 3﷯﷯+ 3﷮ 𝑥 + 4﷯﷯ + 4﷮ 𝑥 + 5﷯﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 3﷯﷮2﷯ 𝑥 + 4﷯﷮3﷯ 𝑥 + 5﷯﷮4 ﷯ 2 𝑥 + 4﷯ 𝑥 + 5﷯ + 3 𝑥 + 3﷯ 𝑥 + 5﷯ + 4 𝑥 + 3﷯ 𝑥 + 4﷯﷮ 𝑥 + 3﷯ 𝑥 + 4﷯ 𝑥 + 5﷯﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 3﷯﷮2﷯ 𝑥 + 4﷯﷮3﷯ 𝑥 + 5﷯﷮4 ﷯﷮ 𝑥 + 3﷯ 𝑥 + 4﷯ 𝑥 + 5﷯﷯ 2 𝑥﷮2﷯+4𝑥+5𝑥+20﷯+3 𝑥﷮2﷯+3𝑥+5𝑥+15﷯+ 4 𝑥﷮2﷯+3𝑥+4𝑥+12﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 3﷯﷮2﷯ 𝑥 + 4﷯﷮2﷯ 𝑥 + 5﷯﷮3 ﷯ 2 𝑥﷮2﷯+9𝑥+20﷯+3 𝑥﷮2﷯+8𝑥+15﷯+4 𝑥﷮2﷯+7𝑥+12﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 3﷯﷮2﷯ 𝑥 + 4﷯﷮2﷯ 𝑥 + 5﷯﷮3 ﷯ 2 𝑥﷮2﷯+18𝑥+40+3 𝑥﷮2﷯+24𝑥+45+4 𝑥﷮2﷯+28𝑥+48﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 3﷯﷮2﷯ 𝑥 + 4﷯﷮2﷯ 𝑥 + 5﷯﷮3 ﷯ 2 𝑥﷮2﷯+3 𝑥﷮2﷯+4 𝑥﷮2﷯18𝑥+24𝑥+28𝑥+40+45+48﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒙 + 𝟑﷯﷮𝟐﷯ 𝒙 + 𝟒﷯﷮𝟐﷯ 𝒙 + 𝟓﷯﷮𝟑 ﷯ 𝟗 𝒙﷮𝟐﷯+𝟕𝟎𝒙+𝟏𝟑𝟑﷯