I believe that the "radius and tangent theorem" is the theorem from geometry that a line tangent to a circle is perpendicular to the radius of the circle to the point of tangency. Since the function is the lower half of the circle , the circle of radius 1 and center (1,7), a point (x,f(x)) is the endpoint of a radius whose other endpoint is the center (1,7), and thus with slope . Therefore the line tangent to our circle at that point is perpendicular to that radius, and so has opposite reciprocal slope; the tangent line at (x,f(x)) has slope
. But the slope of the tangent line is f'(x), so one just plugs into this to get one's answer.
--Kevin C.