Derivative of (x^2+5x-1)/(x^2)

[theBTMANIAC]

Homework Statement

Find the first derivative of [...] (below)

Homework Equations

((x^2)+5x-1)/(x^2)

The Attempt at a Solution

(x^2+5x-1)/(x^2)

y&#039; = \frac{(x^2)(2x+5-0)-((x^2)+5x+-1)(2x)}{x^4}<br /> <br /> \\ <br /> y&#039; = \frac{2x^3+5x^2-2x^3-10x^2+2x}{x^4}<br /> <br /> \\ y&#039; = \frac{(-5x^2)+2x}{x^4}<br /> <br /> \\ y&#039; = \frac{-5x+2}{x^3}<br /> <br />

Is this correct? It feels "off."

 

[matt_crouch]

thats what i got... =]

 

[theBTMANIAC]

Cool. Thank you. :)

 

[HallsofIvy]

However, the simplest way to do that is write
\frac{x^2+ 5x- 1}{x^2}= \frac{x^2}{x^2}+ \frac{5x}{x^2}- \frac{1}{x^2}= 1+ 5x^{-1}- x^{-2}

The derivative of that is, of course, -5x^{-2}+ 2x^{-3}.

Can you see that that is equal to your result?

 

[Anonymous217]

You can also apply the product rule if HallsofIvy's method doesn't work out (for example, if the denominator is more complicated).