Ex 5.5, 6 - Differentiate (x + 1/x)x + x(1 + 1/x) - Logarithmic Differentiation - Type 2

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  1. Class 12
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Ex 5.5,6 (Method 1) Differentiate the functions in, (π‘₯+1/π‘₯)^π‘₯+ π‘₯^((1+1/π‘₯) ) Let 𝑦= (π‘₯+1/π‘₯)^π‘₯+ π‘₯^((1 + 1/π‘₯) ) Let 𝑒 = (π‘₯+1/π‘₯)^π‘₯ , 𝑣 = π‘₯^((1 + 1/π‘₯) ) 𝑦 = 𝑒+𝑣 Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑𝑦/𝑑π‘₯ = (𝑑 (𝑒 + 𝑣))/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑𝑒/𝑑π‘₯ + 𝑑𝑣/𝑑π‘₯ Now 𝑑𝑣/𝑑π‘₯ = 𝑑𝑒/𝑑π‘₯ + 𝑑𝑣/𝑑π‘₯ Putting values of 𝑑𝑒/𝑑π‘₯ & 𝑑𝑣/𝑑π‘₯ π’…π’š/𝒅𝒙 = (𝒙+𝟏/𝒙)^𝒙 ((𝒙^𝟐 βˆ’ 𝟏)/(𝒙^𝟐+ 𝟏)+π₯𝐨𝐠⁑(𝒙+ 𝟏/𝒙) ) + 𝒙^((𝟏 + 𝟏/𝒙) ) ((𝒙 + 𝟏 βˆ’ π’π’π’ˆβ‘π’™ )/𝒙^𝟐 )

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