1. **State the problem:** We have the function $$y = \frac{x^7 + 7x}{7}$$ and need to (a) decompose it into the form $$y = f(u)$$ and $$u = g(x)$$ with non-identity functions, and (b) find $$\frac{dy}{dx}$$ as a function of $$x$$. 2. **Decompose the function:** We can let $$u = x^7 + 7x$$ which is a function of $$x$$. Then, $$y = f(u) = \frac{u}{7}$$. So, $$f(u) = \frac{u}{7}$$ and $$u = g(x) = x^7 + 7x$$. 3. **Find $$\frac{dy}{dx}$$:** Using the chain rule, $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$. 4. Calculate $$\frac{dy}{du}$$: $$\frac{dy}{du} = \frac{d}{du} \left( \frac{u}{7} \right) = \frac{1}{7}$$. 5. Calculate $$\frac{du}{dx}$$: $$\frac{du}{dx} = \frac{d}{dx} (x^7 + 7x) = 7x^6 + 7$$. 6. Combine to get $$\frac{dy}{dx}$$: $$\frac{dy}{dx} = \frac{1}{7} \cdot (7x^6 + 7) = \frac{\cancel{7}x^6 + \cancel{7}}{\cancel{7}} = x^6 + 1$$. **Final answers:** (a) $$f(u) = \frac{u}{7}, \quad u = x^7 + 7x$$ (b) $$\frac{dy}{dx} = x^6 + 1$$