Rewrite the function in a form suitable for differentiation.

The given function is $y = (\sqrt{x}-1)^3$.
We can rewrite $\sqrt{x}$ as $x^{1/2}$.
So, $y = (x^{1/2}-1)^3$.

Let $u = x^{1/2}-1$. Then $y = u^3$.
The chain rule states that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
First, find $\frac{dy}{du}$:

Next, find $\frac{du}{dx}$:

Substitute $u = x^{1/2}-1$ back into $3u^2$:

Now, multiply $\frac{dy}{du}$ and $\frac{du}{dx}$:

Rewrite $x^{1/2}$ as $\sqrt{x}$ and $x^{-1/2}$ as $\frac{1}{\sqrt{x}}$:

The final answer is $\frac{3(\sqrt{x}-1)^2}{2\sqrt{x}}$.