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If ylogx=(x-y) ,prove that dy/dx=logx/(1+logx)^2

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If ylogx=(x-y) ,prove that dy/dx=logx/(1+logx)^2

Posted by N. T. 6 years, 10 months ago

CBSE > Class 12 > Mathematics

1 answers

Prabjeet Singh 6 years, 10 months ago

$\text {Given equation is }y \text { log }x = x-y$ $...(1)$

$\therefore x=y \text { log }x + y$

$\Rightarrow x=y(\text {log }x+1)=y(1 + \text {log } x)$ $...(2)$

$\text {Differentiating Eqn. (1) using product rule, w.r.t. }x, \text { we get,}$

$\cfrac {y}{x} + \text {log } x.\cfrac {dy}{dx} = 1-\cfrac {dy}{dx}$

$\Rightarrow \text {log }x.\cfrac {dy}{dx} + \cfrac {dy}{dx} =1-\cfrac {y}{x}$

$\Rightarrow \cfrac {dy}{dx} \bigg(\text {log }x+1\bigg)=\cfrac {x-y}{x}$

$\Rightarrow \cfrac {dy}{dx}=\cfrac {x-y}{x(1+\text {log }x)}$

$\text {From Eqn. (1) and Eqn. (2),}$

$\Rightarrow \cfrac {dy}{dx} =\cfrac {y \text { log }x}{y(1+\text { log }x)(1+\text {log }x)}$

$\Rightarrow \cfrac{dy}{dx}=\cfrac {\text { log }x}{(1+\text {log }x)^2}$ $\text {Proved}$

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