Draw a vertical line going through point P and consider the triangle made by P, the intersection of the vertical line with the x axis and the intersection of the tangent with the x axis. Then take into account that a tangent has the same derivative as the function at the point where it touches the function (by definition).
I think it's by definition that CP and CQ are the same, line p denotes the circle, which has rise/slide df/dx as sin/cos which is tangent of the angle at P, which is the same angle as the dotted line makes with the x axis. If line p wasn't a radius, the slope would not be tan.