Chapter 3.3, Problem 13E

(a)

To determine

To calculate: The value of $\frac{dy}{dx}$ by applying the Product Rule.

Answer to Problem 13E

The required derivative is $3x^2+2x+1$ .

Explanation of Solution

Given information:

The curve: $y=\left(x+1\right)\left(x^2+1\right)$

Formula used:

Product rule:

  $\frac{d}{dx}\left(uv\right)=u{v}^{\prime }+v{u}^{\prime }$

Calculation:

The given curve is $y=\left(x+1\right)\left(x^2+1\right)$ .

Differentiate the given curve using the product rule $\frac{d}{dx}\left(uv\right)=u{v}^{\prime }+v{u}^{\prime }$ .

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left[\left(x+1\right)\left(x^2+1\right)\right]\\ =\left(x+1\right)\frac{d}{dx}\left(x^2+1\right)+\left(x^2+1\right)\frac{d}{dx}\left(x+1\right)\\ =\left(x+1\right)2x+\left(x^2+1\right)\left(1\right)\\ =2x^2+2x+x^2+1\\ =3x^2+2x+1\end{array}$

Hence, the required derivative is $3x^2+2x+1$ .

(b)

To determine

To calculate: The value of $\frac{dy}{dx}$ by multiplying the factors first and then differentiating.

Answer to Problem 13E

The required horizontal tangents of the curve are $x=0$ , $x=5.052$ and $x=0.197$ .

Explanation of Solution

Given information:

The curve: $y=\left(x+1\right)\left(x^2+1\right)$

Formula used:

Product rule:

  $\frac{d}{dx}\left(uv\right)=u{v}^{\prime }+v{u}^{\prime }$

Calculation:

The given curve is $y=\left(x+1\right)\left(x^2+1\right)$ .

Multiply each term of the given curve with each other.

  $\left(x+1\right)\left(x^2+1\right)=x^3+x^2+x+1$

Differentiate the above expression.

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left(x^3+x^2+x+1\right)\\ =\frac{d}{dx}{x}^3+\frac{d}{dx}{x}^2+\frac{d}{dx}x+\frac{d}{dx}\left(1\right)\\ =3x^2+2x+1+0\\ =3x^2+2x+1\end{array}$

Hence, the required derivative is $3x^2+2x+1$ .