Ex 5.5, 16 - Find derivative of f(x)=(1+x)(1+x2)(1+x4)(1+x8) - Ex 5.5

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.5, 16 Find the derivative of the function given by f (𝑥) = (1 + 𝑥) (1 + 𝑥﷮2﷯) (1 + 𝑥﷮4﷯) (1 + 𝑥8) and hence find f ′(1) . Given 𝑓 𝑥﷯= 1+𝑥﷯ 1+ 𝑥﷮2﷯﷯ 1+ 𝑥﷮4﷯﷯ 1+ 𝑥﷮8﷯﷯ Let 𝑦= 1+𝑥﷯ 1+ 𝑥﷮2﷯﷯ 1+ 𝑥﷮4﷯﷯ 1+ 𝑥﷮8﷯﷯ Taking log both sides log 𝑦 = log 1+𝑥﷯ 1+ 𝑥﷮2﷯﷯ 1+ 𝑥﷮4﷯﷯ 1+ 𝑥﷮8﷯﷯ log 𝑦 = log 1+𝑥﷯+ log﷮ 1+ 𝑥﷮2﷯﷯﷯+ log﷮ 1+ 𝑥﷮4﷯﷯﷯ + log﷮ 1+ 𝑥﷮8﷯﷯﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ = 𝑑 log 1 + 𝑥﷯ + log﷮ 1 + 𝑥﷮2﷯﷯﷯ + log﷮ 1 + 𝑥﷮4﷯﷯﷯+ log﷮ 1 + 𝑥﷮8﷯﷯﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ = 𝑑 log 1 + 𝑥﷯﷯﷮𝑑𝑥﷯ + 𝑑 log﷮ 1 + 𝑥﷮2﷯﷯﷯﷯﷮𝑑𝑥﷯ + 𝑑 log﷮ 1 + 𝑥﷮4﷯﷯﷯﷯﷮𝑑𝑥﷯ + 𝑑 log﷮ 1 + 𝑥﷮8﷯﷯﷯﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮1 + 𝑥﷯ . 𝑑 1 + 𝑥﷯﷮𝑑𝑥﷯ + 1﷮ 1 + 𝑥﷮2﷯﷯﷯ . 𝑑 1 + 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ + 1﷮ 1 + 𝑥﷮4﷯﷯﷯ . 𝑑 1 + 𝑥﷮4﷯﷯﷮𝑑𝑥﷯ + 1﷮ 1 + 𝑥﷮8﷯﷯﷯ . 𝑑 1 + 𝑥﷮8﷯﷯﷮𝑑𝑥﷯ 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮1 + 𝑥﷯ . 0+1﷯ + 1﷮ 1 + 𝑥﷮2﷯﷯﷯ . 0+2𝑥﷯ s + 1﷮ 1 + 𝑥﷮4﷯﷯﷯ . 0+4 𝑥﷮3﷯﷯ + 1﷮ 1 + 𝑥﷮8﷯﷯﷯ . 0+8 𝑥﷮7﷯﷯ 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮1 + 𝑥﷯ + 2𝑥﷮1 + 𝑥﷮2﷯﷯ + 4 𝑥﷮3﷯﷮1 + 𝑥﷮4﷯﷯ + 8 𝑥﷮7﷯﷮1 + 𝑥﷮8﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦 1﷮1 + 𝑥﷯ + 2𝑥﷮1 + 𝑥﷮2﷯﷯ + 4 𝑥﷮3﷯﷮1 + 𝑥﷮4﷯﷯ + 8 𝑥﷮7﷯﷮1 + 𝑥﷮8﷯﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1+𝑥﷯ 1+ 𝑥﷮2﷯﷯ 1+ 𝑥﷮4﷯﷯ 1+ 𝑥﷮8﷯﷯ 1﷮1 + 𝑥﷯ + 2𝑥﷮1 + 𝑥﷮2﷯﷯ + 4 𝑥﷮3﷯﷮1 + 𝑥﷮4﷯﷯ + 8 𝑥﷮7﷯﷮1 + 𝑥﷮8﷯﷯﷯ Hence, 𝒇′ 𝒙﷯ = 𝟏+𝒙﷯ 𝟏+ 𝒙﷮𝟐﷯﷯ 𝟏+ 𝒙﷮𝟒﷯﷯ 𝟏+ 𝒙﷮𝟖﷯﷯ 𝟏﷮𝟏 + 𝒙﷯ + 𝟐𝒙﷮𝟏 + 𝒙﷮𝟐﷯﷯ + 𝟒 𝒙﷮𝟑﷯﷮𝟏 + 𝒙﷮𝟒﷯﷯ + 𝟖 𝒙﷮𝟕﷯﷮𝟏 + 𝒙﷮𝟖﷯﷯﷯ We need to find 𝑓′ 1﷯ Putting 𝑥=1 𝑓′ 1﷯ = 1+1﷯ 1+ 1﷯﷮2﷯﷯ 1+ 1﷯﷮4﷯﷯ 1+ (1)﷮8﷯﷯ 1﷮1 +1﷯ + 2 1﷯﷮1+ 1﷯﷮2﷯﷯ + 4 1﷯﷮3﷯﷮1 + 1﷯﷮4﷯﷯ + 8 1﷯﷮7﷯﷮1 + 1﷯﷮8﷯﷯﷯ = 2 1+1﷯ 1+1﷯ 1+1﷯ 1﷮1 + 1﷯ + 2﷮1 + 1﷯ + 4﷮1 + 1﷯ + 8﷮1 + 1﷯﷯ = 2 2﷯ 2﷯ 2﷯ 1﷮2﷯ + 2﷮2﷯ + 4﷮2﷯ + 8﷮2﷯﷯ = 16 1 + 2 + 4 + 8﷮2﷯﷯ = 16 × 15﷮2﷯ = 8 × 15 = 120

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