1. The problem asks to estimate the derivative $f'(0)$ of the function $f(x)$ at $x=0$. 2. The derivative $f'(x)$ at a point gives the slope of the tangent line to the graph of $f(x)$ at that point. 3. From the graph description, the function $f(x)$ is a downward-opening parabola with vertex at $(0,1)$. 4. The vertex of a parabola is the highest or lowest point, and the slope of the tangent line at the vertex is always zero because the tangent line is horizontal there. 5. Therefore, the slope of the tangent line at $x=0$ is $0$, so $f'(0) = 0$. 6. Among the answer choices, this corresponds to option A: 0.