Here are two geometric ways of thinking about the quotient rule. The first is essentially a geometric interpretation of an algebraic manipulation of the product rule. The second is an interpretation of the quotient rule as it is usually written.

Consider a rectangle with length $x$ and height $y$, with area $A$. We want to determine the change in height $\Delta y$ in response to a change in length $\Delta x$ and/or a change in area $\Delta A$. 

If we hold $x$ constant and increase $A$ by $\Delta A$, then the resulting change in $y$ is $\Delta y = \frac{\Delta A}{x}$. 

[![A drawing of a rectangle with length x, height y, and area A. Another rectangle drawn above the original rectangle represents an increase in area delta-A and an increase in height delta-y.][1]][1]

If we hold $A$ constant and increase $x$ by $\Delta x$, then the resulting decrease in $y$ is $\Delta y \approx - \frac{y\Delta x}{x}$. It's actually $\Delta y = -\frac{(y-\Delta y)\Delta x}{x}$, but we neglect the $\Delta x \Delta y$ term as usual.
[![A drawing of a rectangle with length x, height y, and area A. Another rectangle drawn to the right of the original rectangle shows an increase in area delta-A and an increase in length delta-x. A rectangle drawn on the original rectangle shows a decrease in area minus delta-A and a decrease in height delta-y.][2]][2]

When $A$ and $x$ change simultaneously, a reasonable approximation
for the change in $y$ is
$$\begin{align*}\Delta y &\approx \frac{\Delta A}{x} - \frac{y\Delta x}{x}\\&=\frac{\Delta A}{x} - \frac{A\Delta x}{x^2}\\&=\frac{x\Delta A - A\Delta x}{x^2}.\end{align*}$$

We can interpret this last line geometrically if we extrude our rectangle into a square prism with side lengths $x$ and height $y$. The area of each vertical face is $A=xy$. 

[![A drawing of a rectangular prism with a square base with side lengths x and height y. The resulting increase in volume due to an increase in length delta-x and increase in height delta-y are depicted.][3]][3]

If one side length of the base changes by $\Delta x$, the volume will change by $A\Delta x$. If the height changes by $\Delta y$, the volume will change by $x^2\Delta y$. And if the area of the front face changes by $\Delta A$, then the volume will change by $x\Delta A$.  We can think of the numerator of the quotient rule as representing the relationship between these changes in volume.

[![A drawing of prisms showing that x squared times delta-y is approximately x times delta-A minus A times delta-x.][4]][4]

$$x^2\Delta y \approx x \Delta A - A \Delta x$$

(It's for convenience that we only vary one of the side lengths of the base and hold the other constant. If both are allowed to vary, then we will get the same result after accounting for an additional volume change of $A\Delta x$ in the third dimension.)

  [1]: https://i.sstatic.net/6Hb79m.jpg
  [2]: https://i.sstatic.net/lnCSwm.jpg
  [3]: https://i.sstatic.net/TSxbUm.jpg
  [4]: https://i.sstatic.net/ESfmA.jpg