1. **State the problem:** Differentiate the function $y=3x^3$ and find the slope of the tangent line at the point where $x=1$. 2. **Recall the differentiation rule:** For a function $y = ax^n$, the derivative is given by $$\frac{dy}{dx} = a n x^{n-1}$$ where $a$ and $n$ are constants. 3. **Apply the rule:** Here, $a=3$ and $n=3$, so $$\frac{dy}{dx} = 3 \times 3 x^{3-1} = 9x^2$$ 4. **Evaluate the derivative at $x=1$:** $$\frac{dy}{dx}\bigg|_{x=1} = 9 \times 1^2 = 9$$ 5. **Interpretation:** The slope of the tangent line to the curve $y=3x^3$ at $x=1$ is 9. **Final answer:** The derivative at $x=1$ is $9$.