1. **State the problem:** We are given the implicit equation $$ue^y = 2x + 8y$$ and asked to find $$\frac{dy}{dx}$$. 2. **Recall the formula and rules:** Since $$y$$ is a function of $$x$$, we use implicit differentiation. Differentiate both sides with respect to $$x$$, remembering to apply the chain rule for terms involving $$y$$. 3. **Differentiate both sides:** $$\frac{d}{dx}(ue^y) = \frac{d}{dx}(2x + 8y)$$ 4. **Differentiate left side:** Since $$u$$ is a constant with respect to $$x$$, $$u \frac{d}{dx}(e^y) = u e^y \frac{dy}{dx}$$ by chain rule. 5. **Differentiate right side:** $$\frac{d}{dx}(2x) = 2$$ and $$\frac{d}{dx}(8y) = 8 \frac{dy}{dx}$$. 6. **Write the differentiated equation:** $$u e^y \frac{dy}{dx} = 2 + 8 \frac{dy}{dx}$$ 7. **Group terms with $$\frac{dy}{dx}$$ on one side:** $$u e^y \frac{dy}{dx} - 8 \frac{dy}{dx} = 2$$ 8. **Factor out $$\frac{dy}{dx}$$:** $$\left(u e^y - 8\right) \frac{dy}{dx} = 2$$ 9. **Solve for $$\frac{dy}{dx}$$:** $$\frac{dy}{dx} = \frac{2}{u e^y - 8}$$ 10. **Compare with given options:** None of the options A, B, or C match this derivative exactly, and option D is "هيچيان ن" (none). Therefore, the correct answer is option D. **Final answer:** $$\frac{dy}{dx} = \frac{2}{u e^y - 8}$$ which corresponds to option D (none of the above).