Recall that when a circle completes one full revolution, it moves a distance equal to its circumference, 2πr.

Circle B has a radius of 1 unit while circle C has a radius of 2 units.

So, circle C makes twice as many revolutions as circle B when referenced to circle B (i.e. with respect to circle B).

This means that y/u = 2 (i.e. y is twice of u), or dy/du = 2.

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Similarly, we compare circles A and B.

Circle A has a radius of 3 units while circle B has a radius of 1 unit.

So, circle B makes1/3 times as many revolutions as circle A when referenced to circle A (i.e. with respect to circle A).

This means that u/x = 1/3 (i.e. u is 1/3 of x), or du/dx = 2.

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The system is connected by the common circle B. So, we can say that circle C makes 2/3 as many revolutions as circle A (because when circle B makes 1 revolution, circles C and A make 2 and 3 revolutions respectively).

This means that y/x = 2/3 (i.e. y is 2/3 of x), or dy/dx = 2/3.

This can be seen by the Chain Rule

2/3 = (2) (1/3)
dy/dx = (dy/du) (du/dx)

So, dy/dx = dy/du times du/dx

dy/dx means how many revolutions circle C makes when circle A rotates 1 round.

dy/du means how many revolutions circle C makes when circle B rotates 1 round.

du/dx means how many recolutions circle B makes when circle A rotates 1 round.

dy/dx
= dy/du x du/dx
= 2 x 1/3

(as explained previously - circle C makes twice as many revolutions as circle B which in turn makes 1/3 as many revolutions as circle A)

= 2/3