Seems I can't differentiate properly...or I need some help with it.

rock.freak667

1. The problem statement, all variables and given/known data
Part of a question here...

Show that

[tex]\frac{d}{dx} \left[ \frac{x}{(4-x^2)^n} \right ] = \frac{1-2n}{(4-x^2)^n} - \frac{8n}{(4-x^2)^{n+1}}[/tex]


2. Relevant equations

[tex]\frac{d}{dx}(\frac{u}{v}) = \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} [/tex]

3. The attempt at a solution

So, using the quotient law I get

[tex] \frac{(4-x^2)^n \times 1 -(x) \times n(4-x^2)^{n-1} \times -2x}{(4-x^2)^{2n}}[/tex]

[tex]=\frac{1-2nx^2(4-x^2)^{-1}}{(4-x^2)^n}[/tex]

[tex] =\frac{1}{(4-x^2)^n} - \frac{2nx^2}{(4-x^2)^{n+1}}[/tex]

I seem to have the denominators correct but not the numerators. Did I do it wrong or are there more ways to simplify?

Nick89

I can't get the result either... I get the same as you, only a + instead of a - between the fractions...

EDIT
Maple backs me up... According to maple the result is:
[tex]\frac{1}{(4-x^2)^n} + \frac{2nx^2}{(4-x^2)^{n+1}}[/tex]

And no matter what I do I cannot get it to simplify / expand / factor, whatever, into what you have to show... I don't think they are equal.

CompuChip

I checked it with Mathematica, Nick89 is correct. The thread starter is off by a sign but
[tex]
\frac{d}{dx} \left[ \frac{x}{(4-x^2)^n} \right ] = \frac{1-2n}{(4-x^2)^n} - \frac{8n}{(4-x^2)^{n+1}}
[/tex]
definitely doesn't hold.

Dick

If you are really feeling unconfident that you might be missing a simplification, try putting say x=0 and n=2 and comparing the results of the two. They are different, so your's in right (once you fix that sign error).

rock.freak667

Well I guess I should post the entire question:

[tex]I_n =\int _{0} ^{1} \frac{1}{4-x^2}[/tex]

for n=1,2,3,...
Verify that

[tex]
\frac{d}{dx} \left[ \frac{x}{(4-x^2)^n} \right ] = \frac{1-2n}{(4-x^2)^n} - \frac{8n}{(4-x^2)^{n+1}}[/tex]

and hence prove that

[tex]8I_{n+1}=(2n-1)I_n +\frac{1}{3^n}[/tex]


EDIT: Ahh...-ve*-ve=+ve...Will fix now... but the question being wrong is really odd.

DavidWhitbeck

Rockfreak, look at this, and now take the [tex]x^2[/tex] term on the numerator of the second term and put in

[tex]x^2=-(4-x^2)+4[/tex]

and simplify the result.

rock.freak667

I hate questions where you always need to remember that 1-1=0

DavidWhitbeck

My students absolutely hated those 1/1=1 and 1-1=0 tricks you use to simplify algebraic expressions! There is reason behind them of course, but if done with no explanation they look like magic.

In this one you know that you want to have 4-x^2 on the top of the 2nd term to split it up into the two types of terms that appear in the answer. So you're like I need to express the numerator [tex]x^2[/tex] in the form [tex]a(4-x^2)+b[/tex] for some numbers a and b to get the expression into the form that I want.