Clock: Rates of Change Problem

[Kilo]

Here's the problem:

The minute hand is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o'clock?

It's from my calculus textbook (Stewart)

I want to find how fast the distance changes relative to time, not the angle between the hands.

I decided to use the law of cosines and then take the derivative of it:
d=distance between hands, d'= rate of change of the distance, \theta=angle between hands, \theta'=rate of change of the angle

2dd'=2*4*8*sin(\theta)\theta'

I can find d using law of cosines and angle \theta, but would it be okay to find \theta' by saying:

the hour hand travels \pi/6 per hour and the minute hand travels 2\pi per hour...
\theta'= 2\pi-\pi/6=11\pi/6 per hour

I'm just not sure if that is how I should find \theta'

Thank you!

 

[HallsofIvy]

Yes, that argument is perfectly valid.