1. **Problem:** Find the derivative of $f(x) = (6x^2 + 7x)^4$ using the chain rule. 2. **Formula:** The chain rule states that if $f(x) = (g(x))^n$, then $$f'(x) = n(g(x))^{n-1} \cdot g'(x)$$ 3. **Step 1:** Identify the outer function and inner function: - Outer function: $h(u) = u^4$ - Inner function: $g(x) = 6x^2 + 7x$ 4. **Step 2:** Differentiate the outer function with respect to $u$: $$h'(u) = 4u^3$$ 5. **Step 3:** Differentiate the inner function with respect to $x$: $$g'(x) = \frac{d}{dx}(6x^2 + 7x) = 12x + 7$$ 6. **Step 4:** Apply the chain rule: $$f'(x) = h'(g(x)) \cdot g'(x) = 4(6x^2 + 7x)^3 (12x + 7)$$ 7. **Final answer:** $$\boxed{f'(x) = 4(6x^2 + 7x)^3 (12x + 7)}$$ This derivative shows how the rate of change of the composite function depends on both the outer power and the inner polynomial's slope.