Find the slope of the ray connecting the center of the circle at (0, 0) to the point (x, f(x)) on the circle. Then use the fact that the tangent to a circle is perpendicular to the ray to find the slope of the tangent. Check that it matches the result with the chain rule.
The equation for the top half of a circle is f(x) = √1 - x2. In addition to using implicit differentiation (Example 2.9.10), we can find the slope of the tangent with the chain rule, or find it geometrically.