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

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
\[
\frac{d}{dx}fg=(4x^2+7x+6)(64x-23)+(32x^2-23x+6)(8x+7).
\]
That's it! We can stop here or simplify the derivative further, but we have found the answer that we wanted to find.