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