# Tricky (for me anyway) Differentiation.

• Feb 8th 2009, 01:33 AM
craigmain
Tricky (for me anyway) Differentiation.
Hi

I am trying to differentiate.
Many attempts at trying, and I cannot get to the answer. I am doing something wrong. Just having the answer is not helping. I have tried using the product and quotient rule(s) but am probably not simplifying correctly. If someone can show the steps that would really be great.

$\frac{x^2-1}{\sqrt{x^2+1}}$

Thanks
Regards
Craig.
• Feb 8th 2009, 01:40 AM
mr fantastic
Quote:

Originally Posted by craigmain
Hi

I am trying to differentiate.
Many attempts at trying, and I cannot get to the answer. I am doing something wrong. Just having the answer is not helping. I have tried using the product and quotient rule(s) but am probably not simplifying correctly. If someone can show the steps that would really be great.

$\frac{x^2-1}{\sqrt{x^2+1}}$

Thanks
Regards
Craig.

Use the quotient rule. To get the derivative of the denominator, use the chain rule.

If you post your work it will be easier to know where you're going wrong.
• Feb 8th 2009, 03:15 AM
craigmain
Tricky (for me anyway) Differentiation
Here are some steps.

Here is the start.
$
\frac{2x.\sqrt{x^2+1}+\frac{x.\sqrt{x^2+1}}{(x^2+1 )^2}.(x^2-1)}{(x^2+1)}
$

Then
$
\frac{(x.\sqrt{x^2+1}).(2(x^2+1)^2+(x^2-1))}{(x^2+1)^3}
$

But something is wrong. I need to get to:

$
\frac{x.(x^2+3).\sqrt{x^2+1}}{(x^2+1)^2}
$

I cannot see how simplifying will get the terms I need to cancel.

Cannot seem to get there. I have pages of working, and I am doing something stupid.
• Feb 8th 2009, 03:41 AM
Moo
Hello,
Quote:

Originally Posted by craigmain
Here are some steps.

Here is the start.
$
\frac{2x.\sqrt{x^2+1}+\frac{x.\sqrt{x^2+1}}{(x^2+1 )^2}.(x^2-1)}{(x^2+1)}
$

Then
$
\frac{(x.\sqrt{x^2+1}).(2(x^2+1)^2+(x^2-1))}{(x^2+1)^3}
$

But something is wrong. I need to get to:

$
\frac{x.(x^2+3).\sqrt{x^2+1}}{(x^2+1)^2}
$

I cannot see how simplifying will get the terms I need to cancel.

Cannot seem to get there. I have pages of working, and I am doing something stupid.

You went wrong in the differentiation (1st step)

Quotient rule says :
$\left[\frac uv\right]'=\frac{u'v-uv'}{v^2}$

But what you did is $\frac{u'v-u \left(\frac 1v\right)'}{v^2}$ (Surprised)

Because $-\frac{x\sqrt{x^2+1}}{(x^2+1)^2}$ is the derivative of $\frac{1}{\sqrt{x^2+1}}$, not the derivative of $\sqrt{x^2+1}$
Just a careless mistake, I guess ^^'
• Feb 8th 2009, 03:42 AM
(Hi)
Dont try to jump over the steps(Talking)
I'll give you an example to differentiate

$
\frac{x-1}{\sqrt{x}}
$

After using quotient rule the differntiation becomes

$

\frac{\sqrt{x} \frac{d(x-1)}{dx} - (x-1)\frac{d(\sqrt{x})}{dx} }{\sqrt{x}^2}

$

(Wait)Now try using it in your case and watch the use of quotient rule
• Feb 8th 2009, 03:44 AM
Moo
Quote:

(Hi)
Dont try to jump over the steps(Talking)
I'll give you an example to differentiate

$
\frac{x-1}{\sqrt{x}}
$

After using quotient rule the differntiation becomes

$

\frac{\sqrt{x} \frac{d(x-1)}{dx} - (x-1)\frac{d(\sqrt{x})}{dx} }{\sqrt{x}^2}

$

(Wait)Now try using it in your case and watch the use of quotient rule

He indeed jumped some steps, but he did everything perfectly (Surprised) (or maybe he did more on his paper)
It's just that he didn't apply the formula correctly :P
• Feb 8th 2009, 04:04 AM
Quote:

Originally Posted by craigmain
Here are some steps.

Here is the start.
$
\frac{2x.\sqrt{x^2+1}+\frac{x.\sqrt{x^2+1}}{(x^2+1 )^2}.(x^2-1)}{(x^2+1)}
$

Maybe you are correct (And I I know it will be the case:D)
But
I don't think differentiation of denominator in first case is correct (Thinking) :p
• Feb 8th 2009, 06:48 AM
HallsofIvy
You might find it simpler to write you function as $(x^2- 1)(x^2+ 1)^{-\frac{1}{2}}$ and use the product rule.
• Feb 8th 2009, 07:00 AM
Krizalid
$f(x)=\frac{{{x}^{2}}-1}{\sqrt{{{x}^{2}}+1}}=\frac{{{x}^{2}}+1-2}{\sqrt{{{x}^{2}}+1}}=\sqrt{{{x}^{2}}+1}-2{{\left( {{x}^{2}}+1 \right)}^{-\frac{1}{2}}},$ so

$f'(x)=\frac{x}{\sqrt{{{x}^{2}}+1}}+\frac{2x}{\sqrt {{{\left( {{x}^{2}}+1 \right)}^{3}}}}.$