Chapter 3.3, Problem 35E

To determine

To calculate: The first four derivative of the function.

Answer to Problem 35E

The required first fourth derivative is $24x^{-5}$ .

Explanation of Solution

Given information:

The expression: $y=x^{-1}+x^2$

Formula used:

Power rule of differentiation:

  $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Calculation:

The given expression is $y=x^{-1}+x^2$ .

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

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

Differentiate above derivative.

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

Differentiate the above derivative further.

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

Again, differentiate the above derivative.

  $\begin{array}{c}\frac{d}{dx}\left(\frac{d^{3}y}{d{x}^3}\right)=\frac{d}{dx}\left(-6x^{-4}\right)\\ \frac{d^{4}y}{d{x}^4}=\left(-4\right)\left(-6x^{\left(-4-1\right)}\right)\\ \frac{d^{4}y}{d{x}^4}=24x^{-5}\end{array}$

Hence, the required first fourth derivative is $24x^{-5}$ .