Use the quotient rule to find the derivative of $f(x)/g(x)$ where $f(x)=4x^2+5x-2$ and $g(x)=-3x-7$.

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
$$\frac{df}{dx}=8x+5$$
and
$$\frac{dg}{dx}=-3.$$

Now the quotient rule says that
$$\frac{d}{dx}\left(\frac{f}{g}\right)=\left(g\frac{df}{dx}-f\frac{dg}{dx}\right)\frac{1}{g^2}.$$
Plugging in our functions and derivatives we find that
$$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{(-3x-7)(8x+5)-(4x^2+5x-2)(-3)}{(-3x-7)^2}.$$
That's it! We can stop here or simplify the derivative further, but we have found the answer that we wanted to find.