Use the quotient rule to find the derivative of $f(x)/g(x)$ where $f(x)=32x^2-23x+6$ and $g(x)=4x^2+7x+6$.

To use the quotient rule we first need to find $df/dx$ and $dg/dx$. We found the derivative of a polynomial here so we know that
$$\frac{df}{dx}=64x-23$$
and
$$\frac{dg}{dx}=8x+7.$$

Now the quotient rule says that
$$\frac{d}{dx}\left(\frac{f}{g}\right)=\left(g\frac{df}{dx}-f\frac{dg}{dx}\right)/g^2.$$
Plugging in our functions and derivatives we find that
$$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{(4x^2+7x+6)(64x-23)-(32x^2-23x+6)(8x+7)}{(4x^2+7x+6)^2}.$$
That's it! We can stop here or simplify the derivative further, but we have found the answer that we wanted to find.