Use the product rule to find the derivative of $f(x)g(x)$ where $f(x)=5x^6-16x^5-4x^4+72x^3-3x^2-2x+18$ and $g(x)=42x^2+32x+7$.

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
\[
\frac{df}{dx}=30x^5-80x^4-16x^3+216x^2-6x-2
\]
and
\[
\frac{dg}{dx}=84x+32.
\]

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
\begin{eqnarray}
\frac{d}{dx}fg&=&(42x^2+32x+7)(30x^5-80x^4-16x^3+216x^2-6x-2)\\
&+&(5x^6-16x^5-4x^4+72x^3-3x^2-2x+18)(84x+32)
\end{eqnarray}
That's it! We can stop here or simplify the derivative, but we have found the derivative that we wanted to find.