# Thread: Derivative of rational function

1. ## Derivative of rational function

The question:
Find the derivative of $\displaystyle \frac{t}{\sqrt{t^2 - 4}}$

My attempt:

Using quotient rule:
$\displaystyle \frac{\sqrt{t^2 - 4}(1) - \frac{t^2}{\sqrt{t^2 - 4}}}{t^2 - 4}$

$\displaystyle \frac{1}{\sqrt{t^2 - 4}} - \frac{\frac{t^2}{\sqrt{t^2 - 4}}}{t^2 - 4}$

I'm not sure what to do from here (or whether this is even remotely correct). I know the answer is supposed to be:

$\displaystyle \frac{-4}{(t^2 - 4)^{3/2}}$

Any assistance would be greatly appreciated!

2. Instead of splitting the fraction apart, go back to this step:

$\displaystyle \frac{\sqrt{t^2 - 4}(1) - \frac{t^2}{\sqrt{t^2 - 4}}}{t^2 - 4}$

Multiple both top and bottom by $\displaystyle \sqrt{t^2-4}$ which will rationalize the numerator, giving:

$\displaystyle \frac{(t^2 - 4) - t^2}{(t^2 - 4) \sqrt{t^2 - 4}}$

3. Originally Posted by Glitch
The question:
Find the derivative of $\displaystyle \frac{t}{\sqrt{t^2 - 4}}$

My attempt:

Using quotient rule:
$\displaystyle \frac{\sqrt{t^2 - 4}(1) - \frac{t^2}{\sqrt{t^2 - 4}}}{t^2 - 4}$ Mr F says: This is good. Now multiply the numerator and denominator by $\displaystyle \sqrt{t^2 - 4}$ and then simplify the numerator.

[snip]

I'm not sure what to do from here (or whether this is even remotely correct). I know the answer is supposed to be:

$\displaystyle \frac{-4}{(t^2 - 4)^{3/2}}$

Any assistance would be greatly appreciated!
..

4. Ummm... This is quite discouraging. You should not be forgetting algebra or the addition of fractions. These things should be fixed in your head by the time you get to calculus.

You have $\displaystyle \left({t^{2}}-4\right)^{\frac{3}{2}}$ in the second term.

1) Make it look like that, and
2) Make the denominator of the first term the same and add.

I am tempted to make you do algebra drills for a week after seeing this. You MUST get up to speed. Calculus is far more difficult if you are struggling with algebra.

5. Originally Posted by TKHunny
Ummm... This is quite discouraging. You should not be forgetting algebra or the addition of fractions. These things should be fixed in your head by the time you get to calculus.

You have $\displaystyle \left({t^{2}}-4\right)^{\frac{3}{2}}$ in the second term.

1) Make it look like that, and
2) Make the denominator of the first term the same and add.

I am tempted to make you do algebra drills for a week after seeing this. You MUST get up to speed. Calculus is far more difficult if you are struggling with algebra.

I have great difficulty with math, and I'm really trying hard to set things right. If you feel that I should be doing algebra drills, then I'll do so. If you could be so kind as to direct me to where I could find questions (with solutions) to fulfil this, I'd be grateful.

6. Thank you so much mr fantastic and drumist! I've achieved the correct answer.

7. Originally Posted by Glitch
I have great difficulty with math
There's your trouble, right there. Stop saying that. You must be thinking it, too. Don't.

Any algebra book will help.