# Thread: derivative of a rational function

1. ## Differentiating a Rational Function

Look at the image attached: It shows the differentiation of a rational function using
Wolfram Alpha.
I am having difficulties understanding why the final calculation of the derivative is equal to:
$-\frac{\7}{\ 2(x-2)^2}$

*Note: The final solution is not shown on the image attached.
The entire thing can be viewed here:

http://www.wolframalpha.com/input/?i=Derivative+Calculator&f1=%28x%2B5%29%282x-4%29^-1&f=Derivative.derivativefunction_%28x%2B5%29%28 2x-4%29^-1&a=*FVarOpt-_**Derivative.derivativevariable--

Looking at the steps in the image attached, I understand everything that's happening up until the final line.

From my interpretation it seems it should be:

$\frac {\1(2x-4)-2(x+5)(1)}{\2(x-2)^2}$

= $\frac {2x-4-2x-10}{\(2x-4)^2}$

which:

= $-\frac {\14}{\(2x-4)^2}$

Where is the -7 coming from? What steps am I missing and what am I doing wrong?

2. ## Re: Differentiating a Rational Function

It's algebra.
(2x - 4)^2 = [2(x - 2)]^2 = 4(x - 2)^2
The 4 reduces with the 14.

3. ## Re: Differentiating a Rational Function

Originally Posted by TheChaz
It's algebra.
(2x - 4)^2 = [2(x - 2)]^2 = 4(x - 2)^2
The 4 reduces with the 14.
I see! So simple. By factoring out the two it's simply diving the expression by two. Hence, we divide the numerator by 2 and it equals 7. Thank you so much.

You pointed out exactly what I needed to see. It's funny because I wasn't paying attention to the fact that the expression was modified.
2(x-2)^2

I was immediately interpreting that as (2x-4)^2 which is incorrect. Exponents first!
You knew how I was seeing it the wrong way. Very good insight.