James,
The best way to look at this is that at any given moment, the minute hand and hour hand are at a certain angle apart, creating a triangle with sides 15 and 12mm long. At 2pm, the angle is 2*pi/2/12=pi/3 radians. The distance between the hour and minute hand is the third side of the triangle.
To relate the third side to the first two, use the law of cosines to find the third side (x):
x^2=15^2+12^2-2*15*12*cos theta.
You'll want to plug in theta=pi/3 radians to find x at 2pm.
Now our equation is x^2=339-360 cos theta.
Then differentiate both sides to find dx/dt (the rate the distance is changing). You'll get d theta on the other side:
2x(dx/dt)=360 sin theta (d theta/dt)
d theta/dt can be found from the relative speed between the minute and the hour hand as follows:
The hour hand moves 2*pi radians over 12*60 minutes or pi/360. The minute hand moves 2*pi radians over 60 minutes or pi/30. Since the minute hand is moving faster than the hour hand, both hands are moving at pi/30-pi/360. This is your d theta.
Now plug in all your values to solve for dx/dt. The law of cosines is not often used but very useful when you have two sides of the triangle and need to find the third. Hope that helps!
