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  4. If √[3]y √x = √[6](x+y)5 , then (dy/dx) =

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Q. If $\sqrt[3]{y} \sqrt{x} = \sqrt[6]{\left(x+y\right)^{5}} , $ then $\frac{dy}{dx} = $

KCETKCET 2019Continuity and Differentiability

Solution:

$\sqrt[3]{y}\sqrt{x}=\sqrt[6]{\left(x+y\right)^{5}}$
$\Rightarrow \, y^{2}x^{3}=\left(x+y\right)^{5}$
$\Rightarrow \, y^{2}x^{3}=y^{5}\left(\frac{x}{y}+1\right)^{5}$
$\Rightarrow \, \left(\frac{x}{y}\right)^{3}=\left(\frac{x}{y}+1\right)^{5}$
$\Rightarrow \, \frac{d y}{d x}=\frac{y}{x}$
$f \left(\frac{x}{y}\right)=c$
$\Rightarrow \, \frac{d y}{d x}=\frac{y}{x}$