1. **State the problem:** Differentiate the function $f(x) = (6e^{2x} - 3)^3$ with respect to $x$. 2. **Recall the chain rule:** If $y = [u(x)]^n$, then $\frac{dy}{dx} = n[u(x)]^{n-1} \cdot \frac{du}{dx}$. 3. **Identify inner function:** Let $u(x) = 6e^{2x} - 3$. 4. **Differentiate inner function:** $$\frac{du}{dx} = 6 \cdot \frac{d}{dx} e^{2x} - 0 = 6 \cdot 2e^{2x} = 12e^{2x}$$ 5. **Apply chain rule:** $$\frac{df}{dx} = 3(6e^{2x} - 3)^2 \cdot 12e^{2x}$$ 6. **Simplify:** $$\frac{df}{dx} = 36e^{2x}(6e^{2x} - 3)^2$$ **Final answer:** $$\boxed{\frac{df}{dx} = 36e^{2x}(6e^{2x} - 3)^2}$$