1. **State the problem:** Find the derivative of the function $f(x) = x(x-7)^7$. 2. **Formula and rules:** We will use the product rule for derivatives, which states: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $u(x) = x$ and $v(x) = (x-7)^7$. 3. **Find derivatives of each part:** - Derivative of $u(x) = x$ is $u'(x) = 1$. - Derivative of $v(x) = (x-7)^7$ uses the chain rule: $$v'(x) = 7(x-7)^6 \cdot 1 = 7(x-7)^6$$ 4. **Apply the product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = 1 \cdot (x-7)^7 + x \cdot 7(x-7)^6$$ 5. **Factor common terms:** $$f'(x) = (x-7)^6 \big((x-7) + 7x\big) = (x-7)^6 (x - 7 + 7x) = (x-7)^6 (8x - 7)$$ **Final answer:** $$f'(x) = (x-7)^6 (8x - 7)$$