Chapter 3.1, Problem 34E

The graph of $y={\left(x^2-1\right)}^4{\left(x^2+1\right)}^5$ is shown in Fig. 3. Find the coordinates of the local maxima and minima. [Hint: Use Example 4.]

Example 4

Differentiating and Simplifying

Find

$\frac{dy}{dx}$, where $y={\left(x^2-1\right)}^4{\left(x^2+1\right)}^5$.

Solution

Let $f\left(x\right)={\left(x^2-1\right)}^4$, $g\left(x\right)={\left(x^2+1\right)}^5$, and use the product rule. The general power rule is needed to compute $f\text{'}(x)$ and $g\text{'}(x)$:

$\frac{dy}{dx}={\left(x^2-1\right)}^4\cdot \frac{d}{dx}{\left(x^2+1\right)}^5+{\left(x^2+1\right)}^5\cdot \frac{d}{dx}{\left(x^2-1\right)}^4$

$={\left(x^2-1\right)}^4\cdot 5{\left(x^2+1\right)}^4\cdot \left(2x\right)+{\left(x^2+1\right)}^5\cdot 4{\left(x^2-1\right)}^3\cdot \left(2x\right)$