Chapter 2.3, Problem 34E

To determine

To calculate: The derivative $\frac{dy}{dx}$ of the function

Answer to Problem 34E

The derivative of the function is. $y^{\prime }=2x+1$ , $y^{″}=2$ , $y^{‴}=0$ , $y^4=0$ .

Explanation of Solution

Given information:

The function is $y=x^2+x+3$ .

Concept used:

The formula used is $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ .

Calculation:

The function is $y=x^2+x+3$ .

Differentiate the function with respect to $x$ for the first order derivative.

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

Differentiate the function with respect to $x$ for the second order derivative.

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

Differentiate the function with respect to $x$ for the third order derivative.

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

Differentiate the function with respect to $x$ for the fourth order derivative.

  $\begin{array}{c}\frac{d}{dx}\left(\frac{d^{3}y}{d{x}^4}\right)=\frac{d}{dx}\left(0\right)\\ =0\end{array}$

Conclusion: The derivative of the function is. $y^{\prime }=2x+1$ , $y^{″}=2$ , $y^{‴}=0$ , $y^4=0$ .