A clock's minute hand has length 4 and its hour hand length 3.

What is the distance between the tips at the moment when it is increasing most rapidly?

The fact that both hands are moving is irrelevant. In other words the answer is the same if you say decreasing instead of increasing. You have a triangle with two known side lengths. The question is at what angle between the know sides would changing that same angle by a fixed amount cause the greatest change in the length of the third unknown side. The answer is when the shorter known side is at 90° to the unknown side. If the known sides are equal the answer is 0. With sides of 3 and 4 the answer is the square root of (4²-3²) or 2.64575