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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 49 of 72
Marks: +1, -0
1+1x1−1x\frac{1+\frac{1}{x}}{1-\frac{1}{x}}
Solution:  
Let f (x) = 1+1x1−1x\frac{1+\frac{1}{x}}{1-\frac{1}{x}} ... (i)
Differentiating (i) with respect to x, we get
ddx\frac{d}{dx} (f (x)) =
(1−1x)(1+1x)−(1+1x)(1−1x)(1−1x)2\frac{\left(1-\frac{1}{x}\right)\left(1+\frac{1}{x}\right)-\left(1+\frac{1}{x}\right)\left(1-\frac{1}{x}\right)}{\left(1-\frac{1}{x}\right)^{2}}
=
(1−1x)(−1x2)−(1+1x)(1x2)(1−1x)2\frac{\left(1-\frac{1}{x}\right)\left(-\frac{1}{x^{2}}\right)-\left(1+\frac{1}{x}\right)\left(\frac{1}{x^{2}}\right)}{\left(1-\frac{1}{x}\right)^{2}}
= −1x2+1x3−1x2−1x3(1−1x)2\frac{-\frac{1}{x^{2}}+\frac{1}{x^{3}}-\frac{1}{x^{2}}-\frac{1}{x^{3}}}{\left(1-\frac{1}{x}\right)^{2}}
= −2x2(1−1x)2\frac{-\frac{2}{x^{2}}}{\left(1-\frac{1}{x}\right)^{2}}
= −2x2(x−1)2x2\frac{-\frac{2}{x^{2}}}{\frac{(x-1)^{2}}{x^{2}}} = −2(x−1)2\frac{-2}{(x-1)^{2}} , x ≠ 0 , 1
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