Chapter 3.3, Problem 33E

To determine

To calculate: The first four derivative of the function.

Answer to Problem 33E

The required first fourth derivative is 24.

Explanation of Solution

Given information:

The expression: $y=x^4+x^3-2x^2+x-5$

Formula used:

Power rule of differentiation:

  $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Calculation:

The given expression is $y=x^4+x^3-2x^2+x-5$ .

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

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left(x^4+x^3-2x^2+x-5\right)\\ \frac{dy}{dx}=\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(x^3\right)-\frac{d}{dx}\left(2x^2\right)+\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(5\right)\\ \frac{dy}{dx}=4x^{\left(4-1\right)}+3x^{\left(3-1\right)}-4x^{\left(2-1\right)}+\left(1\right)x^{\left(1-1\right)}-0\\ \frac{dy}{dx}=4x^3+3x^2-4x+1\end{array}$

Differentiate above derivative.

  $\begin{array}{c}\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(4x^3+3x^2-4x+1\right)\\ \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(3x^2\right)-\frac{d}{dx}\left(4x\right)+\frac{d}{dx}\left(1\right)\\ \frac{d^{2}y}{d{x}^2}=3\left(4x^{\left(3-1\right)}\right)+2\left(3x^{\left(2-1\right)}\right)-1\left(4x^{\left(1-1\right)}\right)+0\\ \frac{d^{2}y}{d{x}^2}=12x^2+6x-4\end{array}$

Differentiate the above derivative further.

  $\begin{array}{c}\frac{d}{dx}\left(\frac{d^{2}y}{d{x}^2}\right)=\frac{d}{dx}\left(12x^2+6x-4\right)\\ \frac{d^{3}y}{d{x}^3}=\frac{d}{dx}\left(12x^2\right)+\frac{d}{dx}\left(6x\right)-\frac{d}{dx}\left(4\right)\\ \frac{d^{3}y}{d{x}^3}=2\left(12x^{\left(2-1\right)}\right)+1\left(6x^{\left(1-1\right)}\right)-0\\ \frac{d^{3}y}{d{x}^3}=24x+6\end{array}$

Again, differentiate the above derivative.

  $\begin{array}{c}\frac{d}{dx}\left(\frac{d^{3}y}{d{x}^3}\right)=\frac{d}{dx}\left(24x+6\right)\\ \frac{d^{4}y}{d{x}^4}=\frac{d}{dx}\left(24x\right)+\frac{d}{dx}\left(6\right)\\ \frac{d^{4}y}{d{x}^4}=24+0\\ \frac{d^{4}y}{d{x}^4}=24\end{array}$

Hence, the required first fourth derivative is 24.