# Differential

• May 9th 2006, 03:56 AM
Bert
Differential
Hello,

I have here a strength differential $\frac{-t}{\sqrt{1-t^2}}$

Normally is not a big problem but I making a mistake I don't know:

let my try $u=1-t^2$ thus $u'=-2t$

$v=\sqrt{u}$ thus $v'=\frac{1}{2}u^{-\frac{1}{2}$

And $w=\frac{-t}{v}$ thus $w'=\frac{-v+tv'}{v^2}$

Now I need to multiply al the thinks and then I find $\frac{t}{1-t^2}$ but it's not correct.

Where do I miss ? Greets.
• May 9th 2006, 04:33 AM
CaptainBlack
Quote:

Originally Posted by Bert
Hello,

I have here a strength differential $\frac{-t}{\sqrt{1-t^2}}$

Normally is not a big problem but I making a mistake I don't know:

let my try $u=1-t^2$ thus $u'=-2t$

$v=\sqrt{u}$ thus $v'=\frac{1}{2}u^{-\frac{1}{2}$

And $w=\frac{-t}{v}$ thus $w'=\frac{-v+tv'}{v^2}$

Now I need to multiply al the thinks and then I find $\frac{t}{1-t^2}$ but it's not correct.

Where do I miss ? Greets.

Stage 1 - Product rule:

$
\frac{d}{dt}\left(\frac{-t}{\sqrt{1-t^2}}\right)=\frac{1}{\sqrt{1-t^2}}\frac{d}{dt} (-t)+(-t)\frac{d}{dt} \ \left(\frac{1}{\sqrt{1-t^2}}\right)
$

$
=-\frac{1}{\sqrt{1-t^2}}-t\ \frac{d}{dt} \ \left(\frac{1}{\sqrt{1-t^2}}\right)
$

Stage 2 - Chain rule on last term on the right:

$
=-\frac{1}{\sqrt{1-t^2}}-t\ (-1/2)\frac{1}{(1-t^2)^{3/2}}\frac{d}{dt}(1-t^2)
$

$
=\frac{1}{\sqrt{1-t^2}}-t\ (-1/2)\frac{1}{(1-t^2)^{3/2}}(-2t)
$

$
=\frac{1}{\sqrt{1-t^2}}-\frac{t^2}{(1-t^2)^{3/2}}
$

From here you should be able to simplify this yourself.

RonL
• May 10th 2006, 12:47 AM
Bert
thus you can splits this in two parts by the produc rule

then you have $\frac{1}{\sqrt{1-t^2}}$

I can find then on this way: $u=1-t^2$ Thus $u'=-2t$

Then $v=\sqrt{u}$ and $v'=\frac{1}{2}u^{-\frac{1}{2}}$

Then $\frac{\frac{-t}{\sqrt{1-t^2}}}{\sqrt{1-t^2}^2}=\frac{-t}{\sqrt{1-t^2}}\frac{1}{1-t^2}$

But why is this fault ? $u=1-t^2 \rightarrow u'=-2t \ \ \ v=\sqrt{u} \rightarrow\frac{1}{2}u^{-\frac{1}{2}} \ \ \ w=\frac{1}{v} \rightarrow w'=lnv$

And then multiply them what is mis? Greets.
• May 10th 2006, 02:16 PM
ThePerfectHacker
Quote:

Originally Posted by Bert
But why is this fault ? $u=1-t^2 \rightarrow u'=-2t \ \ \ v=\sqrt{u} \rightarrow\frac{1}{2}u^{-\frac{1}{2}} \ \ \ w=\frac{1}{v} \rightarrow w'=lnv$

And then multiply them what is mis? Greets.

Mistake that if,
$w=\frac{1}{v}$
then,
$w'=-\frac{1}{v^2}$
You confused with anti-derivative.
• May 11th 2006, 12:06 AM
Bert
of cours thank you very well.

a small problem If I simplify $\frac{-t}{\sqrt{1-t^2}}\frac{1}{1-t^2}
$

then I get $-\frac{t}{{(1-t^2)}^{\frac{3}{2}}}}$ And this it not the same as I differentiate the original by my computer program then I get $-\frac{1}{{(1-t^2)}^\frac{3}{2}}$
• May 11th 2006, 06:55 AM
earboth
Quote:

Originally Posted by Bert
thus you can splits this in two parts by the produc rule
then you have $\frac{1}{\sqrt{1-t^2}}$
I can find then on this way: $u=1-t^2$ Thus $u'=-2t$

Then $v=\sqrt{u}$ and $v'=\frac{1}{2}u^{-\frac{1}{2}}$

...

Hello,

Then $v=\sqrt{u}$ and $v'=\frac{1}{2}u^{-\frac{1}{2}}\ \gets \mbox{this looks funny to me}$

If you use the chain rule on v' then you get: $v'=\frac{1}{2}u^{-\frac{1}{2}} \cdot (-2t)=-t \cdot u^{-\frac{1}{2}}$

Greetings

EB
• May 11th 2006, 07:16 AM
Bert
Thank you very well I find them great good forum :) Greets Thanks