Stuck on radical rational derivative problem for 10 hours.

[jddoxtator]

I have been stuck on this problem for over 10 hours now.

I can't seem to be able to reduce the radical rational expression down to the answer.
I know that the answer is -(1/2)a^-3/2, but I am getting completely lost in the factoring and there is no example of the factoring online.
The answer online says to use the Power Rule, but at this point in the textbook the Power Rule has not been taught yet and it is expected that you are to use the #4 definition above.

I have a feeling it is rustiness with radical expressions that is hanging me up, but I have looked at all my textbooks on radical expressions, watched a plethora of videos about radical expressions and nothing is getting me closer to solving this problem.

I get stuck mostly when all I can to is flip conjugates from numerator and denominator over and over again.
I am clearly missing something, but I don't know what.
If someone could post the factoring for this it would be GREATLY appreciated.
Seeing the process will help me see where I am going wrong and solidify the rule sets being used.

This is about as far as I get before things go completely off the rails.
Heck, this may already be completely off the rails if i am missing something fundamental.

 

[lookagain]

The conjugate of \(\displaystyle \ (\sqrt{a} - \sqrt{a + h}) \ \) is \(\displaystyle \ (\sqrt{a} + \sqrt{a+h}). \) You need the numerator
and the denominator multiplied at the same time by this conjugate.

You're missing grouping symbols for the fractional exponent for the answer:
-(1/2)a^(-3/2)

 

[jddoxtator]

The conjugate of \(\displaystyle \ (\sqrt{a} - \sqrt{a + h}) \ \) is \(\displaystyle \ (\sqrt{a} + \sqrt{a+h}). \) You need the numerator
and the denominator multiplied at the same time by this conjugate.

You're missing grouping symbols for the fractional exponent for the answer:
-(1/2)a^(-3/2)

I am aware that this is the form of a conjugate, I just have not applied any in my work example.
I have factored it to where this is the next step that needs to be applied to demonstrate that this is where things start to go wrong for me.
Or are you saying I should be performing the conjugate before I even consider the denominator (h) in the formula?

Here is the example worked a bit further with a conjugate.
As you can see, you quickly reach a point where another conjugate is required and this seems to just continue endlessly after this point.
It feels like an infinite loop, that is the problem I am having.
I can see where the -(1/2) is starting to form, but where the a^(-3/2) comes from is beyond where I have reached yet.
Either I am stopping too soon, or I am missing some fundamental factoring technique or understanding.

 

[Dr.Peterson]

Or are you saying I should be performing the conjugate before I even consider the denominator (h) in the formula?

Yes. What you did, multiplying by the product of the radicals, will not help. Multiplying by the conjugate will.

See example 3 here for a problem using this technique:

Calculus I - The Definition of the Derivative

In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function.

tutorial.math.lamar.edu

So just do this instead of what you did:

​

 

[jddoxtator]

Alright, I think i finally have the solution.
The link to Paul's Online Notes was helpful, as I figured out that I had to use a combination of the techniques in Example 2 and Example 3.
This is what i came up with.

 

[Dr.Peterson]

Excellent.

I'll just make a trivial comment:

​

Once you've replaced h with 0 (which you could do because the remaining expression was continuous), you are no longer taking a limit; you've taken it! Now you're just simplifying the answer.

The usual error students make is to write "lim" too few times, rather than too many; that's an actual mistake. Yours is only a slight waste of ink.

 

[jddoxtator]

Excellent.

I'll just make a trivial comment:

View attachment 39030​

Once you've replaced h with 0 (which you could do because the remaining expression was continuous), you are no longer taking a limit; you've taken it! Now you're just simplifying the answer.

The usual error students make is to write "lim" too few times, rather than too many; that's an actual mistake. Yours is only a slight waste of ink.

Ah, keep to the definition of the task then I suppose.
Indeed, ink is now over $20 a bottle!
This was a frustrating experience, but it strengthened my knowledge of radicals in rational expressions significantly.
I think I will make more use of this site "Paul's Online Notes" when I find myself stuck again.
Better than trying to drag net youtube and my old textbooks.

 

[Dr.Peterson]

This was a frustrating experience, but it strengthened my knowledge of radicals in rational expressions significantly.

Yes, I often tell students that calculus is where you finally really learn algebra, because you have to use it regularly as a tool.

I think I will make more use of this site "Paul's Online Notes" when I find myself stuck again.

I've been recommending it for decades, I think! Anything textbook-y will generally help more than random searches. And that one even has a link back to his algebra notes.

 

[khansaheb]

View attachment 39023
View attachment 39024
I have been stuck on this problem for over 10 hours now.

I can't seem to be able to reduce the radical rational expression down to the answer.
I know that the answer is -(1/2)a^-3/2, but I am getting completely lost in the factoring and there is no example of the factoring online.
The answer online says to use the Power Rule, but at this point in the textbook the Power Rule has not been taught yet and it is expected that you are to use the #4 definition above.

I have a feeling it is rustiness with radical expressions that is hanging me up, but I have looked at all my textbooks on radical expressions, watched a plethora of videos about radical expressions and nothing is getting me closer to solving this problem.

I get stuck mostly when all I can to is flip conjugates from numerator and denominator over and over again.
I am clearly missing something, but I don't know what.
If someone could post the factoring for this it would be GREATLY appreciated.
Seeing the process will help me see where I am going wrong and solidify the rule sets being used.

This is about as far as I get before things go completely off the rails.
Heck, this may already be completely off the rails if i am missing something fundamental. View attachment 39025

A well known result from the given definition is:

\(\displaystyle \frac{d}{dx} x^n = n* x^{(n-1)} \)

Could you have used the result above?

 

[Dr.Peterson]

A well known result from the given definition is:

\(\displaystyle \frac{d}{dx} x^n = n* x^{(n-1)} \)

Could you have used the result above?

No:

The answer online says to use the Power Rule, but at this point in the textbook the Power Rule has not been taught yet and it is expected that you are to use the #4 definition above.