Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 21

Show that your answers agree in finding the derivative of a function $y = (x^2 + 1)(x^3 + 1)$ in two ways: by using the Product Rule and by performing the multiplication first.

Solution:

First, by using Product Rule we get,


$
\begin{equation}
\begin{aligned}

y' =& (x^2 + 1) \frac{d}{dx} (x^3 + 1) + (x^3 + 1) \frac{d}{dx} (x^2 + 1)\\
\\
y' =& (x^2 + 1)(3x^2) + (x^3 + 1)(2x)\\
\\
y' =& 3x^4 + 3x^2 + 2x^4 + 2x\\
\\
y' =& 5x^4 + 3x^2 + 2x

\end{aligned}
\end{equation}
$


Last, by performing multiplication or FOIL method


$
\begin{equation}
\begin{aligned}

y =& x^5 + x^2 + x^3 + 1\\
\\
y' =& \frac{d}{dx} (x^5) + \frac{d}{dx} (x^2) + \frac{d}{dx} (x^3) + \frac{d}{dx} (1)\\
\\
y' =& 5x^4 + 2x + 3x^2 + 0\\
\\
y' =& 5x^4 + 3x^2 + 2x

\end{aligned}
\end{equation}
$


By using the two different methods, we can say that our answers agree.