Taking the derivative of a polynomial fraction?

1. Sep 26, 2012

mooha

Taking the derivative of a polynomial fraction??

b]1. The problem statement, all variables and given/known data[/b]
Ok, so the question wants me to differentiate f(x)= (x)/(x+1). We are supposed to use the definition of the derivative f'(x)= (limit as h->0) [f(x+h)-f(x)]/(h). We also have learned the power rule. I did the formula and had some issues, so I tried to check my answer with the power rule. However they didn't come out the same. I'm not sure where I made the mistake, the formula or the power rule!

2. Relevant equations[/b
f'(x)= (limit as h->0) [f(x+h)-f(x)]/(h)

3. The attempt at a solution
Here's my work:

f'(x)= (lim h->0) [(x+h)/(x+h+1) - x/(x-1)] / h

=(lim h->0) [(x+h)/(x+h+1) - x/(x-1)]/h * (x+h+1)(x+1)

= (lim h->0) [(x+h)(x+1) - (x)(x+h+1)] / (h)(x+h+1)(x+1)

= (lim h->0) (x^2+x+hx+h-x^2-xh-x)/(xh+h^2x+hx^2+h+h^2+hx)

= (lim h->0) h/(h)(x+hx+x^2+h+x)

=(lim h->0) 1/x^2+2x+hx+h

f'(x)= 1/x^2+2x

Power Rule:

y'= (x)/(x+1)
y'= (x)*(x+1)^-1
y'= -(x+1)^-2
y'= -1/(x+1)^2

Thank you so much!!

2. Sep 26, 2012

SammyS

Staff Emeritus
Re: Taking the derivative of a polynomial fraction??

Hello mooha. Welcome to PF !

It's pretty difficult to read line after line of fractions in the form of
[(x+h)(x+1) - (x)(x+h+1)] / (h)(x+h+1)(x+1), etc,​
especially when you don't always use proper parentheses, as in
1/x^2+2x+hx+h, which I assume you intended to be $\displaystyle \frac{1}{x^2+2x+hx+h}\ .$​

You might make you life easier by writing f(x) as
$\displaystyle f(x)=1-\frac{1}{x+1}\ .$​

3. Sep 26, 2012

mooha

Re: Taking the derivative of a polynomial fraction??

Unfortunately my computer does not have an easy way to do that! Unless you are able to do that with the thread options? It would make my life a lot easier! It is hard to keep putting in those parenthesis! :) Thanks

4. Sep 26, 2012

SammyS

Staff Emeritus
Re: Taking the derivative of a polynomial fraction??

I will look at it more carefully now.

5. Sep 26, 2012

SammyS

Staff Emeritus
Re: Taking the derivative of a polynomial fraction??

You don't need to expand the denominator, & apparently that's where your mistake came from.

6. Sep 26, 2012

mooha

Re: Taking the derivative of a polynomial fraction??

Ok thank you! So my use of the power rule is correct?

7. Sep 26, 2012

mooha

Re: Taking the derivative of a polynomial fraction??

I fixed my mistake, but the answer from the power rule does not match up with the one from the equation.
WAIT! does the power rule go to zero?? because it is (x)(x+1)-1 and the x goes to zero? does the power rule not work at all for this kind of function?

8. Sep 26, 2012

SammyS

Staff Emeritus
Re: Taking the derivative of a polynomial fraction??

Yes the power rule works for (x)(x+1)-1 ...

after you use the product rule !

Try the alternate expression for f(x): f(x) = 1 - (x+1)-1 .

9. Sep 26, 2012

Ray Vickson

Re: Taking the derivative of a polynomial fraction??

It is hard to read such expressions without parentheses, so I don't bother trying.

RGV