# (x+5)/(2x-4) Finding derivative question?

Hi, I was supposed to find the derivative of g(x)=(x+5)/(2x-4) and I got a totally different answer from the one that my friend's answer. Please explain why g’(x)=(2x-4)/(2x-4)^2 +(-2(x+5))/(2x-4)^2

would become g’(x)=-7/(2(2x-4)^2 ) instead of g’(x)=-14/(2x-4)^2 ). Also, is my graph correct? THX!

### 1 Answer

- ?Lv 77 years agoFavorite Answer
Regardless of how your friend did it or how you did it, i'm still going to provide the full step by step solution.

For simplicity let g(x) = y.

Then y = (x + 5) / (2x - 4)

instead of quotient rule i'd use logarithmic differentiation.

ln y = ln [ (x + 5) / (2x - 4) ]

ln y = ln(x + 5) - ln(2x - 4)

y ' / y = (1 / (x + 5) ) - 2 / (2x - 4)

y' = y [ (1 / (x + 5) ) - 2 / (2x - 4) ]

we now back substitute y,

y ' = [ (x + 5) / (2x - 4) ] [ (1 / (x + 5) ) - 2 / (2x - 4) ]

y ' = [1 /(2x - 4) ] -2(x+5) / (2x-4)^2

y ' = [ 2x - 4 - 2(x+5) ] / (2x - 4)^2

y ' = (2x - 4 -2x - 10) / (2x - 4)^2

y ' = -14 / (2x - 4)^2

And this is the answer.

So you did get it CORRECT.

Done!

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Notice that -7/(2(2x-4)^2 ) = -14/(2x-4)^2

They may look different but they are identical.

Subtracting the 2 results in zero -

http://www.wolframalpha.com/input/?i=simplify++-7%...

Wolfram chooses LHS as derivative. I choose RHS.

They are equivalent.

In algebra you can write an equation in countless number of ways.

consider this simple expression x. You can write x thousands of different ways --

1) x^2 / x

2) (8x + x^2) / (8 + x)

3) (37x^5 /x^4 ) - 36x

These all are indeed x. Likewise there can be millions of more ways to write x.

So wolfram sometimes messes up with writing it differently. However it is the same thing.

The derivative per wolfram is

http://www.wolframalpha.com/input/?i=derivative+%2...

but it is same as my answer. Just written in another manner. Why wolfram does this, i have no idea.