We are given $y = \sin^2(x^3)$. We need to find $\frac{dy}{dx}$. We will use the chain rule.

First, differentiate $\sin^2(x^3)$ with respect to $\sin(x^3)$. $\frac{d}{d(\sin(x^3))} (\sin^2(x^3)) = 2\sin(x^3)$

Next, differentiate $\sin(x^3)$ with respect to $x^3$. $\frac{d}{d(x^3)} (\sin(x^3)) = \cos(x^3)$

Finally, differentiate $x^3$ with respect to $x$. $\frac{d}{dx} (x^3) = 3x^2$

Now, multiply all the derivatives together: $\frac{dy}{dx} = 2\sin(x^3) \cdot \cos(x^3) \cdot 3x^2 = 6x^2\sin(x^3)\cos(x^3)$