Answer
${dy}=(\frac{1}{3\sqrt x(1+\sqrt x)^2}){dx}$
Work Step by Step
We evaluate the function: $y=\frac{2\sqrt x}{3{(1+\sqrt x)}}$
on differentiating the above:
$\frac{dy}{dx}=\frac{d\frac{2\sqrt x}{3{(1+\sqrt x)}}}{dx}$
or ${dy}=(\frac{{3{(1+\sqrt x)}}\frac{2}{2\sqrt {x}})-{\frac{6\sqrt x}{2\sqrt x}}}{9(1+\sqrt x)^2}){dx}$
or ${dy}=(\frac{1+\sqrt x-\sqrt x}{3\sqrt x(1+\sqrt x)^2}){dx}$
or ${dy}=(\frac{1}{3\sqrt x(1+\sqrt x)^2}){dx}$
The final answer is: ${dy}=(\frac{1}{3\sqrt x(1+\sqrt x)^2}){dx}$
