1. **State the problem:** Differentiate the function $$f(x) = (x+1)^6$$ with respect to $$x$$. 2. **Formula used:** Use the chain rule for differentiation. If $$f(x) = [g(x)]^n$$, then $$f'(x) = n[g(x)]^{n-1} \cdot g'(x)$$. 3. **Apply the chain rule:** Here, $$g(x) = x+1$$ and $$n=6$$. 4. **Calculate derivative of inner function:** $$g'(x) = \frac{d}{dx}(x+1) = 1$$. 5. **Combine results:** $$f'(x) = 6(x+1)^{5} \cdot 1 = 6(x+1)^5$$. 6. **Final answer:** The derivative of $$f(x) = (x+1)^6$$ is $$f'(x) = 6(x+1)^5$$.