# If y=x^(1/2)+1/x^(1/2) the prove 2x dy/dx+y=2.x^(1/2)

Solution: We  have , $y= \sqrt{x}+ \frac{1}{\sqrt{x}}$              [ differentiate both side w.r.t x  ]

$\frac{dy}{dx}=\frac{1}{2\sqrt{x}}- \frac{1}{2x \sqrt{{x}}}$

put the value of   $\frac{dy}{dx}$   in the equation $2x\frac{dy}{dx}+y=2\sqrt{x}$                  $\because y=\sqrt{x}+\frac{1}{\sqrt{x}}$

$= 2x(\frac{1}{2\sqrt{x}}-\frac{1}{2x\sqrt{x}})+\sqrt{x}+\frac{1}{\sqrt{x}}$

$= (\sqrt{x} -\frac{1}{\sqrt{x}})+\sqrt{x}+\frac{1}{\sqrt{x}}=2\sqrt{x}$

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