1. We are given the function $f(x) = (x^2 + 1)^2$ and asked to find the derivative at $x=2$, i.e., $f'(2)$. 2. To find $f'(x)$, we use the chain rule. If $f(x) = [g(x)]^2$ where $g(x) = x^2 + 1$, then $$f'(x) = 2g(x) \cdot g'(x)$$ 3. First, compute $g'(x)$: $$g'(x) = \frac{d}{dx}(x^2 + 1) = 2x$$ 4. Substitute back into the derivative formula: $$f'(x) = 2(x^2 + 1) \cdot 2x = 4x(x^2 + 1)$$ 5. Now evaluate at $x=2$: $$f'(2) = 4 \cdot 2 \cdot (2^2 + 1) = 8 \cdot (4 + 1) = 8 \cdot 5 = 40$$ 6. Therefore, the derivative of $f(x)$ at $x=2$ is $40$.