1. **State the problem:** Differentiate the function $$y = \frac{e^{-4x}}{4 e^{4x}}$$ with respect to $$x$$. 2. **Simplify the function:** $$y = \frac{e^{-4x}}{4 e^{4x}} = \frac{1}{4} \cdot \frac{e^{-4x}}{e^{4x}} = \frac{1}{4} e^{-4x - 4x} = \frac{1}{4} e^{-8x}$$ 3. **Differentiate using the chain rule:** The derivative of $$e^{u}$$ with respect to $$x$$ is $$e^{u} \cdot \frac{du}{dx}$$. Here, $$u = -8x$$, so $$\frac{du}{dx} = -8$$. 4. **Calculate the derivative:** $$\frac{dy}{dx} = \frac{1}{4} \cdot e^{-8x} \cdot (-8) = -2 e^{-8x}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = -2 e^{-8x}}$$