Calculus · Hard problem

A security camera tilts down by $\theta(d)=\arctan\!\left(\dfrac{100}{d}\right)$ radians to track a person a horizontal distance $d$ meters away. At $d=100$, the tilt rate is $\theta'(100)$. Find the distance $d$ at which the tilt rate has slowed to exactly half of $\theta'(100)$.