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$$f(x) = \frac{x-1}{x+1}$$

From the Definition I have this so far. I am stuck and do not know how to continue.

$$\begin{align} Q(h) &= \frac{f(h)-f(2)}{h} \\&= \frac{ \frac{h-1}{h+1} - \frac{1}{3} }{h} \end{align}$$

Thanks in advance!

Victoria
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    (f(2+h)-f(2))/h, not (f(h)-f(2))/h. – Did Apr 14 '14 at 17:17
  • I've edited your post to show how how to format your equations with fractions, and how to line up several equations in a row. If you edit your post, you'll see how it works. –  Apr 14 '14 at 17:19

1 Answers1

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A this point, you should forget that you're doing a problem about derivatives, and think back to your high school algebra class.

How do you compute with fractions? You know how to add them, subtract them, multiply them, divide them, and simplify them. Do that as best you can, and then go back to remembering this is a problem about derivatives, and see what you can do at that point.

For example, recall that to add two fractions, you can find a common denominator:

$$ \frac{p}{q} + \frac{r}{s} = \frac{ps}{qs} + \frac{qr}{qs} = \frac{ps+qr}{qs}$$

(incidentally, you forgot the limit part when you swapped in the definition of the derivative)

(edit: if you grind this through, you will find that the $h$ doesn't cancel, and the limit will not exist. In problems where we expect derivatives to exist, this usually indicates that you made an error along the way -- and Did points out in the comments that you did indeed make a mistake)

  • Thanks, my biggest problem is actually figuring out the Algebra. When I use what Did wrote I still do not understand how to cancel out h. – Victoria Apr 14 '14 at 17:25
  • Then show what you wrote using my hint--this way we can help and spot where you made a mistake, if you did. – Did Apr 14 '14 at 17:29
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    In particular, it's quite possible you actually know what you're doing, but just made an arithmetic error; @Did asks that you show the work so we can tell what's up. One trick for verifying your work in simplifying a fraction with variables in it is to plug in values; e.g. go through all of your work and plug in $h=1$ and see if you get the same value at every step. If the value of the fraction changes, that's where you made an algebra error! –  Apr 14 '14 at 17:31
  • After simplifying I get (4h^2 + 2h)/(3h^3 + 9h^2) is that possible? – Victoria Apr 14 '14 at 18:07
  • @Victoria Show. Your. Work. In. Details. – Did Apr 14 '14 at 18:30
  • @Did

    ((2+h-1/2+h+1) - 1/3)/h = ((2+h-1/2+h+1) - 1/3) * (1/h) = (2+h-1/2h+h^2+h) - 1/3h = ((3h(2+h-1))/3h(2h+h^2+h)) - (2h+h^2+h/3h(2h+h^2+h))

     – Victoria Apr 14 '14 at 19:36
  • Can't read. Add this to your question and do the simplifications, if any. – Did Apr 14 '14 at 19:40
  • This is so frustrating, I keep getting these unreadable results and I dont even know how to type them in here to show my work! Please help! – Victoria Apr 15 '14 at 13:59
  • There is a comment on this page which answers exactly that: "I've edited your post to show [you] how to format your equations with fractions, and how to line up several equations in a row. If you edit your post, you'll see how it works." – Did Apr 16 '14 at 19:35