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.