1. Problem: Given the function $$y = (-3 - 3x^7)^5$$, we need to find its derivative $$f'(x)$$. 2. Use the chain rule to find the derivative. The outer function is $$u^5$$ where $$u = -3 - 3x^7$$. 3. First, find the derivative of the outer function with respect to $$u$$: $$\frac{d}{du} (u^5) = 5u^4$$. 4. Next, find the derivative of the inner function $$u = -3 - 3x^7$$ with respect to $$x$$: $$\frac{du}{dx} = 0 - 3 \cdot 7x^{6} = -21x^{6}$$. 5. By the chain rule, the derivative $$f'(x)$$ is: $$f'(x) = 5u^4 \cdot \frac{du}{dx} = 5(-3 - 3x^7)^4 \cdot (-21x^6)$$. 6. Simplify the expression: $$f'(x) = -105x^{6}(-3 - 3x^{7})^{4}$$. Final answer: $$\boxed{f'(x) = -105x^{6}(-3 - 3x^{7})^{4}}$$