Let's use different representation for intervals, namely $$(x,\Delta x)$$, $$\Delta x\geq 0$$, which corresponds to interval that spans from $$x-\Delta x$$ to $$x+\Delta x$$. Note that we can multiply intervals by real numbers, e.g. $$c(x, \Delta x)=(cx,|c|\Delta x)$$. When $$x>0$$, $$y>0$$, $$\Delta x\ll x$$ and $$\Delta y\ll y$$, we have the following formulae: $$(x,\Delta x)+(y,\Delta y)=(x+y,\Delta x+\Delta y)$$ $$(x,\Delta x)-(y,\Delta y)=(x-y,\Delta x+\Delta y)$$ $$(x,\Delta x)\cdot(y,\Delta y)=(xy,x\Delta y+y\Delta x)$$ $$(x,\Delta x)/(y,\Delta y)=(xy,x\Delta y+y\Delta x)/y^2$$. So, for multiplication and division the width of the resulting interval depends on $$x$$ and $$y$$, not only on $$\Delta x$$, $$\Delta y$$.