Chapter 3.3, Problem 34E

To determine

To calculate: The first four derivative of the function.

Answer to Problem 34E

The required first fourth derivative is 0.

Explanation of Solution

Given information:

The expression: $y=x^2+x+3$

Formula used:

Power rule of differentiation:

  $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Calculation:

The given expression is $y=x^2+x+3$ .

To find the value of the given expression, use the power rule of differentiation $\frac{d}{dx}{x}^n=n{x}^{n-1}$ .

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

Differentiate above derivative.

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

Differentiate the above derivative further.

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

Again, differentiate the above derivative.

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

Hence, the required first fourth derivative is 0.