Chapter 3.1, Problem 36E

To determine

To find:

The differential equation of the given equation

Answer to Problem 36E

The equation is $f^{\prime }\left(x\right)=1-\frac{1}{x^2}$

Explanation of Solution

Given:

The function is

  $f\left(x\right)=x+\frac{1}{x}$

Concept used:

Definition of the differentiation:-Differentiation is the action of computing a derivative

The derivative of a function $y=f\left(x\right)$ of a variable $x$ is a measure of the rate at which the value y of the function changes with respect to $x$

Calculation:

The function

  $f\left(x\right)=x+\frac{1}{x}\mathrm{...................}\left(1\right)$

The derivative of a function

  $\begin{array}{l}y=f\left(x\right)\\ y^{\prime }=f^{\prime }\left(x\right)=\frac{dy}{dx}\end{array}$

Differentiating the equation (1) with respect to $x$

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

Draw the table

  $y=f\left(x\right)=x+\frac{1}{x}$

Test one point in each of the region formed by the graph

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function

$x-axis$	$1$	$-1$	$0$	$0.382$
$y-axis$	$2$	$-2$	$3$	$3$