Chapter 3.3, Problem 18E

To determine

To calculate: The value of $\frac{dy}{dx}$ .

Answer to Problem 18E

The required value of $\frac{dy}{dx}$ is $-\frac{5}{x^2}+\frac{2}{x^3}$

Explanation of Solution

Given information:

The expression: $y=\frac{x^2+5x-1}{x^2}$

Formula used:

Power rule:

  $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Calculation:

The given expression is $y=\frac{x^2+5x-1}{x^2}$ .

Rewrite the above expression as:

  $\begin{array}{c}y=\frac{x^2+5x-1}{x^2}\\ =\frac{\cancel{x^2}}{\cancel{x^2}}+\frac{5x}{x^2}-\frac{1}{x^2}\\ =1+\frac{5}{x}-\frac{1}{x^2}\\ =1+5x^{-1}-x^{-2}\end{array}$

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

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

Simplify the expression further.

  $\begin{array}{c}\frac{dy}{dx}=-5x^{-2}+2x^{-3}\\ \frac{dy}{dx}=-\frac{5}{x^2}+\frac{2}{x^3}\end{array}$

Hence, the required value of $\frac{dy}{dx}$ is $-\frac{5}{x^2}+\frac{2}{x^3}$ .