Robotic Kinematics & Angular Velocity Mapping (Mechanical Engineering)

A mechanical automation engineer is programming a high-speed robotic sorting arm on an industrial manufacturing assembly line. The base of the arm pivots at an angle $\theta$ (in radians) to track an item moving along a parallel linear conveyor belt located at a fixed perpendicular clearance of $1\text{ meter}$ from the pivot.

The item moves along the belt such that its horizontal position coordinate $x$ (in meters) changes over time $t$ (in seconds) according to:

The rotational tracking angle $\theta$ required to maintain alignment with the item is modeled by:

1. Express the angular velocity $\frac{d\theta}{dt}$ using the Chain Rule in terms of $\frac{d\theta}{dx}$ and $\frac{dx}{dt}$.
2. Differentiate the outer inverse trigonometric function with respect to $x$.
3. Differentiate the rational conveyor position function $x(t)$ with respect to $t$ using the Quotient Rule.
4. Determine the position $x(1)$ and the required angular tracking velocity of the robotic actuator motor ($\frac{d\theta}{dt}$) at $t = 1\text{ second}$.