1. **State the problem:** We are given the function $$y = \left(\frac{x}{7} + \frac{7}{x}\right)^7$$ and asked to: (a) Decompose it into the form $$y = f(u)$$ and $$u = g(x)$$ with non-identity functions. (b) Find the derivative $$\frac{dy}{dx}$$ as a function of $$x$$. 2. **Decomposition:** Let $$u = g(x) = \frac{x}{7} + \frac{7}{x}$$. Then $$y = f(u) = u^7$$. 3. **Find $$\frac{dy}{dx}$$ using the chain rule:** The chain rule states: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$. 4. **Calculate $$\frac{dy}{du}$$:** Since $$y = u^7$$, $$\frac{dy}{du} = 7u^6$$. 5. **Calculate $$\frac{du}{dx}$$:** Given $$u = \frac{x}{7} + \frac{7}{x}$$, $$\frac{du}{dx} = \frac{d}{dx}\left(\frac{x}{7}\right) + \frac{d}{dx}\left(\frac{7}{x}\right) = \frac{1}{7} - \frac{7}{x^2}$$. 6. **Combine to find $$\frac{dy}{dx}$$:** $$\frac{dy}{dx} = 7u^6 \left(\frac{1}{7} - \frac{7}{x^2}\right)$$. 7. **Substitute back $$u$$:** $$\frac{dy}{dx} = 7\left(\frac{x}{7} + \frac{7}{x}\right)^6 \left(\frac{1}{7} - \frac{7}{x^2}\right)$$. 8. **Simplify the factor inside parentheses:** $$\frac{1}{7} - \frac{7}{x^2} = \frac{x^2 - 49}{7x^2}$$. 9. **Final expression:** $$\frac{dy}{dx} = 7\left(\frac{x}{7} + \frac{7}{x}\right)^6 \cdot \frac{x^2 - 49}{7x^2}$$. 10. **Cancel the 7 in numerator and denominator:** $$\frac{dy}{dx} = \left(\frac{x}{7} + \frac{7}{x}\right)^6 \cdot \frac{x^2 - 49}{x^2}$$. **Answer:** (a) $$\{f(u), u\} = \{u^7, \frac{x}{7} + \frac{7}{x}\}$$ (b) $$\frac{dy}{dx} = \left(\frac{x}{7} + \frac{7}{x}\right)^6 \cdot \frac{x^2 - 49}{x^2}$$