What exactly is your problem? I got the same result for x(t). The angular speed is (7/2) * (2π radians) / (1 min) = 7π radians/min. Therefore, the angle the radius to point A makes at time t is 7πt radians. Next, x(t) is the sum of the projection of A, i.e., 2cos(7πt), and the projection of the rod, which can be found using Pythagoras theorem (the y-coordinate of A is 2sin(7πt)).
The derivative can be found using the chain rule. It's a big expression, but sin(7πt) can be factored out. One can show that the second factor cannot equal zero. Therefore, x'(t) = 0 iff sin(7πt) = 0.