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 $\Delta y = \frac{\Delta A}{x}$.

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.

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 $A=xy$.

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.

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

(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.)

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 $\Delta y = \frac{\Delta A}{x}$.

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.

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 $A=xy$. Let's consider the changes in volume that result from changing $x$, $y$, and $A$ on the front face.

If we fix the height and vary the length of the front face by $\Delta x$, the volume of the prism will increase by $A\Delta x$. If we fix the length of the front face and vary the height by $\Delta y$, the volume will increase by $x^2\Delta y$. Changes in $x$ and $y$ will result in some change in the area of the front face. If we are given that the area of the front face changes by $\Delta A$, then the change in the volume of the prism is $x\Delta A$. We can think of the numerator of the quotient rule as representing the relationship between these changes in volume.

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

(It's for convenience that we only vary the length of the front face and hold the length of the side face 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.)