1. **State the problem:** We need to find the derivative of the function $$y=(1+5x-x^2)^{\frac{1}{4}}$$ and then evaluate it at $$x=5$$. 2. **Recall the formula:** For a function of the form $$y=[u(x)]^n$$, the derivative is given by the chain rule: $$\frac{dy}{dx} = n[u(x)]^{n-1} \cdot \frac{du}{dx}$$ 3. **Identify components:** Here, $$u(x) = 1+5x-x^2$$ and $$n=\frac{1}{4}$$. 4. **Compute $$\frac{du}{dx}$$:** $$\frac{du}{dx} = 5 - 2x$$ 5. **Apply the chain rule:** $$\frac{dy}{dx} = \frac{1}{4}(1+5x-x^2)^{\frac{1}{4}-1} \cdot (5 - 2x) = \frac{1}{4}(1+5x-x^2)^{-\frac{3}{4}} (5 - 2x)$$ 6. **Evaluate at $$x=5$$:** Calculate the inside function: $$1 + 5(5) - 5^2 = 1 + 25 - 25 = 1$$ Calculate the derivative factor: $$5 - 2(5) = 5 - 10 = -5$$ 7. **Substitute values:** $$\frac{dy}{dx}\bigg|_{x=5} = \frac{1}{4} \cdot 1^{-\frac{3}{4}} \cdot (-5) = \frac{1}{4} \cdot 1 \cdot (-5) = -\frac{5}{4}$$ **Final answer:** $$\boxed{-\frac{5}{4}}$$