# Thread: Ffinding the difference quotient of x + (4/x).

1. ## Ffinding the difference quotient of x + (4/x).

Given x + (4/x) finding the difference quotient I am wondering how to get started.

I don't know why but, for the life of me I can't clear the fractions and I keep thinking no matter what I do i'm not going to end up with like denominators.

I have gotten use to doing difference quotient by itself.. but, given whole's and fractions here my head is sorta spinning... I haven't missed the D.Q. much at all lately thats for sure.

2. Originally Posted by Chimera
Given x + (4/x) finding the difference quotient I am wondering how to get started.

I don't know why but, for the life of me I can't clear the fractions and I keep thinking no matter what I do i'm not going to end up with like denominators.

I have gotten use to doing difference quotient by itself.. but, given whole's and fractions here my head is sorta spinning... I haven't missed the D.Q. much at all lately thats for sure.
Aren't we just doing

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

where $\,f(x) = x+\frac{4}{x}$? Where are you stuck?

3. I am trying to clear the top ((x+h) + (x/4) - (x + (4/x) and I am not quite sure how to go about doing this because, I keep thinking if I multiply by (x + h) it is going to complicate the fraction more than it should.. and still not get like denominators for (4/x) and (4/(x+h))...

4. Originally Posted by Chimera
I am trying to clear the top ((x+h) + (x/4) - (x + (4/x) and I am not quite sure how to go about doing this because, I keep thinking if I multiply by (x + h) it is going to complicate the fraction more than it should.. and still not get like denominators for (4/x) and (4/(x+h))...
$\displaystyle \frac{\Delta f(x)}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$

$\displaystyle =\frac{x+h+\frac{4}{x+h}-(x+\frac{4}{x})}{h}$

$\displaystyle =\frac{h+\frac{4}{x+h}-\frac{4}{x}}{h}$

$\displaystyle =\frac{h+\frac{4x-4(x+h)}{x(x+h)}}{h}$

$\displaystyle =\frac{h+\frac{-4h}{x(x+h)}}{h}$

$\displaystyle =\frac{h+\frac{-4h}{x(x+h)}}{h}$

$\displaystyle = 1+\frac{-4}{x(x+h)}$

$\displaystyle = \frac{x^2+hx-4}{x(x+h)}$

I did this somewhat hastily and don't have time to check the work.

5. I'm not sure if it is just my instructors preference or if there are more "rules" regarding it off the top of my head.

However, I know it has always been emphasized to me to never cancel out a letter when it is being added on the top of the fraction. The only time we have been allowed to cancel is when a number is being factored out.

ex. x(x+h)/x

well you can cancel x because its being multiplied. 1 * 1 = 1.so it would be the same as 1(x+)/1 where as with your addition.. I am not saying it is incorrect.. however, it is a bit harder to follow.

6. Originally Posted by Chimera
I'm not sure if it is just my instructors preference or if there are more "rules" regarding it off the top of my head.

However, I know it has always been emphasized to me to never cancel out a letter when it is being added on the top of the fraction. The only time we have been allowed to cancel is when a number is being factored out.

ex. x(x+h)/x

well you can cancel x because its being multiplied. 1 * 1 = 1.so it would be the same as 1(x+)/1 where as with your addition.. I am not saying it is incorrect.. however, it is a bit harder to follow.
I do not know what you mean by "cancel out a letter when it is being added on the top of the fraction."

7. you canceled out h+(fract) over h by rewriting it as 1 + canceling h off the bottom and top however, last I checked you could not cancel h out like that because it was being added.. you could only cancel in fractions if something was being multiplied not if it was being added/subtracted.

8. $\displaystyle =\frac{h\left(1+\frac{-4}{x(x+h)}\right)}{h}$

$\displaystyle = 1+\frac{-4}{x(x+h)}$

better?