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If $x=e^{\Large \frac{x}{y}},prove\; that \; \large {\frac{dy}{dx}=\frac{x-y}{xlog x}}$

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Toolbox:
  • A function $f(x,y)$ is said to be implicit if it is jumbled in such a way,that it is not possible to write $y$ exclusively s a function of $x$.
  • $\large\frac{d}{dx}$$\phi(y)=\large\frac{d}{dy}$$\phi(y).\large\frac{dy}{dx}$
Step 1:
$x=e^{\Large\frac{x}{y}}$
This can be written as
$\log x=\large\frac{x}{y}\Rightarrow $$y=\large\frac{x}{\log x}$
Differentiating on both sides w.r.t $x$ (Apply quotient rule)
$\large\frac{1}{x}$$=\large\frac{y.1-x dy/dx}{y^2}$
$\Rightarrow y^2=xy-x^2dy/dx$
Therefore $x^2\large\frac{dy}{dx}=$$xy-y^2$
$\Rightarrow \large\frac{dy}{dx}=\frac{y(x-y)}{x^2}$
Step 2:
Substituting $y=\large\frac{x}{\log x}$
$\large\frac{dy}{dx}=\frac{x(x-y)}{\log x.x^2}$
$\Rightarrow \large\frac{dy}{dx}=\frac{(x-y)}{x\log x}$
Hence proved.
answered Jul 2, 2013 by sreemathi.v
 
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