Answer
$\frac{dy}{dx}=4\sqrt{(x+\sqrt{x})}+2x\frac{{({1+\frac{1}{2\sqrt{x}}})}}{\sqrt{(x+\sqrt{x})}}$
Work Step by Step
$y =4x\sqrt{(x+\sqrt{x})}$
on differentiating both sides:
$\frac{dy}{dx}=4\sqrt{(x+\sqrt{x})}\frac{dx}{dx}+4x\frac{d(\sqrt{(x+\sqrt{x})})}{dx}$
$\frac{dy}{dx}=4\sqrt{(x+\sqrt{x})}+4x\frac{1}{2\sqrt{(x+\sqrt{x})}}\frac{d({(x+\sqrt{x})})}{dx}$
$\frac{dy}{dx}=4\sqrt{(x+\sqrt{x})}+4x\frac{1}{2\sqrt{(x+\sqrt{x})}}{({1+\frac{1}{2\sqrt{x}}})}$
$\frac{dy}{dx}=4\sqrt{(x+\sqrt{x})}+2x\frac{{({1+\frac{1}{2\sqrt{x}}})}}{\sqrt{(x+\sqrt{x})}}$
