Differentiation Quotient rule

There is a question I am getting answers for but, when requires to differentiate it again to find a stationary point, it doesn't really work out well, so any help would be great:

Differentiate: (x+3)/(x^2+9)^1/2

Thanks

Is that the specific question - find any stationary points of the function?

tbh it's just an algebra slog - you'll need to post some working if you want someone to check it.

One thing to note is that if you need a fraction to be 0 then the numerator has to be 0, so if you clear any fractions (i.e. negative indices) that occur in the numerator you can pretty much ignore the denominator in your calculations.

The stationary points are the second part, I just need help with the differentiation here is what I have done, but not sure if the answer is right:
v'=1/2*2x*(x^2+9)=x/((x^2+9)^1/2)
u'=1

So using the quotient rule:

((x^2+9)^1/2*1 - x/((x^2+9)^1/2)*(x+3))/x^2+9

=((x+3)-(x^2+3x)/((x^2+9)^1/2))/x^2+9

That's what I have so far, but the two fractions on top of each other has got me into a bit of a mess with the algebra, which is what I need help with,

Thanks

The stationary points are the second part, I just need help with the differentiation here is what I have done, but not sure if the answer is right:
v'=1/2*2x*(x^2+9)=x/((x^2+9)^1/2)
u'=1

So using the quotient rule:

((x^2+9)^1/2*1 - x/((x^2+9)^1/2)*(x+3))/x^2+9

=((x+3)-(x^2+3x)/((x^2+9)^1/2))/x^2+9

That's what I have so far, but the two fractions on top of each other has got me into a bit of a mess with the algebra, which is what I need help with,

Thanks

urgh

LaTex would be nice

Are you differentiating

$\dfrac{x+3}{\sqrt{x^2+9}}$

I am not sure why you have a fraction over a fraction - but I cannot really tell what you have


Looking again at your differentiation - you need to either

Just put numerator = 0

OR

multiply through by $\sqrt{x^2+9}$

I haven't checked your algebra (about to go offline!), but if what you've done so far is correct, then just multiply top and bottom by (x^2 + 9)^1/2 and then set the numerator equal to 0 - see if it leads you to the correct answer (do you have the answer(s) for this question?)

urgh

LaTex would be nice

Are you differentiating

$\dfrac{x+3}{\sqrt{x^2+9}}$

I am not sure why you have a fraction over a fraction - but I cannot really tell what you have


Looking again at your differentiation - you need to either

Just put numerator = 0

OR

multiply through by $\sqrt{x^2+9}$


DO you have $\sqrt{x^2+9}-\frac{x^2+3x}{\sqrt{x^2+9}}$ as your numerator?

Yes I am differentiating that, when you say multiply through by that, is that after using the quotient rule, or should I do it before hand?

after

Is it possible for you to show the algebra leading up to this, because I think I went wrong somewhere?

Well, I wouldn't bother - I would just put the numerator = 0

From the beginning? so we just ignore the denominator?

No, once you have differentiated

Once I differentiate, I get a fraction on top a fraction as shown above, I get the answer ( sorry I can't do LaTex right now):

((x+3)-(x^2+3x)/((x^2+9)^1/2))/x^2+9

This was through the Quotient rule, I think if you do it through the quotient rule, you would get this also, because the derivative of v ( (x^2+9)^1/2 is a fraction due to the negative power,
so am I equating the numerator of this to zero now and ignoring the x^2+9 ?

Simplify your expression, you should just get a nice:

$\dfrac{dy}{dx} = \dfrac{3(3-x)}{(x^2+9)^{\frac{3}{2}}}$.

All you have to do then is equate $3(3-x)$ to zero.

Thanks, is that all I have to do- that's the differentiated expression?

Yes, that's the expression you already had but simplified, look to see how you can rearrange yours to get that.

Once I differentiate, I get a fraction on top a fraction as shown above, I get the answer ( sorry I can't do LaTex right now):

((x+3)-(x^2+3x)/((x^2+9)^1/2))/x^2+9

This was through the Quotient rule, I think if you do it through the quotient rule, you would get this also, because the derivative of v ( (x^2+9)^1/2 is a fraction due to the negative power,
so am I equating the numerator of this to zero now and ignoring the x^2+9 ?

So your numerator is what I posted in post 4

Set this = 0 and solve

Numerator is all that matters since 0/something = 0