Find derivative of: (3 - x^2)/(4 + x^2); Quotient Rule vs. Product Rule

Click For Summary

Homework Help Overview

The discussion revolves around finding the derivative of the function (3 - x^2)/(4 + x^2) using both the Quotient Rule and the Product Rule. Participants are exploring the differences in their results and the potential algebraic mistakes made while applying these rules.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • The original poster attempts to differentiate the function using both the Quotient Rule and the Product Rule, noting discrepancies in their results. Some participants question the algebraic steps taken, particularly regarding the manipulation of fractions and the squaring of terms. Others suggest reviewing the application of the rules to identify where the mistakes may have occurred.

Discussion Status

Participants are actively engaging with the original poster's attempts, providing feedback on the algebraic reasoning and clarifying misconceptions about combining fractions. There is a focus on understanding the correct application of differentiation rules without reaching a definitive conclusion on the original poster's method.

Contextual Notes

There is an emphasis on the importance of correctly applying algebraic operations when working with derivatives, particularly in the context of combining fractions and ensuring that operations do not alter the values incorrectly. The original poster expresses confusion about cancellation rules in algebra.

Lo.Lee.Ta.
Messages
217
Reaction score
0
Hi, I got the right answer when I used the Quotient Rule but not when I used the Product Rule...
I think it might be an algebra mistake...

Product Rule Method:

f'(x) = (3 - x^2)*(4 + x^2)^-1

= (3 - x^2)[(-1(4 + x^2)^-2)*2x] + [(4 + x^2)^-1](-2x)

= [(3 - x^2)(-2x)]/[(4 + x^2)^2] + [(-2x)/(4 + x^2)]

To get the same denominator here, I thought I might square the factor on the right.

= (-6x + 2x^3 + 4x^2)/[(4 + x^2)^2] <----- This is not right.


THE RIGHT ANSWER IS: -14x/[(4 + x^2)^2]
I found that by using the Quotient Rule.
...But where am I going wrong with the Product Rule...?


P.S. Oh, and a side note:
If I wrote out the Product Rule answer like this: (-6x + 2x^3 + 4x^2)/(x^4 + 8x^2 + 16)

Would the 4x^2 and the 8x^2 be able to cancel somewhat?
Sometimes I get confused about what can cancel.
Could it turn into:
(-6x + 2x^3)/(x^4 + 2 + 16) = (-6x + 2x^3)/(x^4 + 18)?

I know that's not the right answer, but is this how it would cancel?

Thank you very much!
 
Physics news on Phys.org
given a fraction like (a+b / (a + c) you can't cancel the a terms out to get b / c

but if you have a * b / a * c then you can cancel the a factors out to get b / c

with that in mind try reviewing your work to see where you went wrong, the product and quotient rules should give the same answers.
 
rewritten a bit more correctly what you surely meant
Lo.Lee.Ta. said:
Hi, I got the right answer when I used the Quotient Rule but not when I used the Product Rule...
I think it might be an algebra mistake...

Product Rule Method:

f(x) = (3 - x^2)*(4 + x^2)^-1

f'(x) = (3 - x^2)[(-1(4 + x^2)^-2)*2x] + [(4 + x^2)^-1](-2x)

= [(3 - x^2)(-2x)]/[(4 + x^2)^2] + [(-2x)/(4 + x^2)]

To get the same denominator here, I thought I might square the factor on the right.

= (-6x + 2x^3 + 4x^2)/[(4 + x^2)^2] <----- This is not right.
As far as I understand what you are saying, there is just no rationale for your operation of squaring. You have a sum, represent it X + Y . You are saying if you square a bit of it -> X + Y2 that will be the same thing?

You have something of form A/B2 + C/B .

In general how do you combine that into one?
 
Lo.Lee.Ta. said:
Hi, I got the right answer when I used the Quotient Rule but not when I used the Product Rule...
I think it might be an algebra mistake...

Product Rule Method:

f'(x) = (3 - x^2)*(4 + x^2)^-1
Be careful to write what you mean! This is f, not f'. You have not yet differentiated.
f(x)= (3- x^2)(4+ x^2)^{-1}

f'(x)= (3 - x^2)[(-1(4 + x^2)^-2)*2x] + [(4 + x^2)^-1](-2x)
Okay, this is correct.

= [(3 - x^2)(-2x)]/[(4 + x^2)^2] + [(-2x)/(4 + x^2)]

To get the same denominator here, I thought I might square the factor on the right.
No. Squaring a number changes the number. Multiply the numerator and denominator by 4+ x^2

= (-6x + 2x^3 + 4x^2)/[(4 + x^2)^2] <----- This is not right.
Yes, it is not right. If you have 2/3 and want to make the denominator 9, you multiply both numerator and denominator to get 6/9 which, of course, represents the same number as 2/3. If you square, you get 4/9 which is NOT the same.

Going back to
-\frac{2x(3- x^2)}{(4+x^2)^2}- \frac{2x}{4+ x^2}
if you multipy both numerator and denominator of the second fraction by 4+ x^2 you get
-\frac{6x- 2x^3}{(4+ x^2)^2}- \frac{8x+ 2x^3}{(4+ x^2)^2}
= \frac{-6x+ 2x^3- 8x- 2x^3}{(4+ x^2)^2}= \frac{-14x}{(4+x^2)^2}

THE RIGHT ANSWER IS: -14x/[(4 + x^2)^2]
I found that by using the Quotient Rule.
...But where am I going wrong with the Product Rule...?P.S. Oh, and a side note:
If I wrote out the Product Rule answer like this: (-6x + 2x^3 + 4x^2)/(x^4 + 8x^2 + 16)

Would the 4x^2 and the 8x^2 be able to cancel somewhat?
Sometimes I get confused about what can cancel.
Could it turn into:
(-6x + 2x^3)/(x^4 + 2 + 16) = (-6x + 2x^3)/(x^4 + 18)?

I know that's not the right answer, but is this how it would cancel?

Thank you very much!
 
Last edited by a moderator:
WOW! HallsofIvy! You gave me the BEST answer I have ever gotten in my experience on Physics Forums!

You wrote out EXACTLY where I went wrong, and explained WHY it was wrong!

I get it now! My squaring method was off!

I worked the problem by multiplying instead by (4 + x^2) and got the right answer!

Thank you SO much for your help!
With the other answers, I didn't understand what was wrong with my method!

If this were YahooAnswers, you'd get voted "Best Answer!" :)
 
Lo.Lee.Ta. said:
WOW! HallsofIvy! You gave me the BEST answer I have ever gotten in my experience on Physics Forums!

You wrote out EXACTLY where I went wrong, and explained WHY it was wrong!

I get it now! My squaring method was off!

I worked the problem by multiplying instead by (4 + x^2) and got the right answer!

Thank you SO much for your help!
With the other answers, I didn't understand what was wrong with my method!

If this were YahooAnswers, you'd get voted "Best Answer!" :)

Just apply what you've learned: \frac{(3-x^{2})&#039;(4+x^{2})-(4+x^{2})&#039;(3-x^{2})}{(4+x^{2})^{2}}=\frac{-14x}{(x^{2}+4)^{2}}
 
Lo.Lee.Ta. said:
WOW! HallsofIvy! You gave me the BEST answer I have ever gotten in my experience on Physics Forums!

You wrote out EXACTLY where I went wrong, and explained WHY it was wrong!

I get it now! My squaring method was off!

I worked the problem by multiplying instead by (4 + x^2) and got the right answer!

Thank you SO much for your help!
With the other answers, I didn't understand what was wrong with my method!

If this were YahooAnswers, you'd get voted "Best Answer!" :)

You heard it here first. :-p

epenguin said:
rewritten a bit more correctly what you surely meant

...

As far as I understand what you are saying, there is just no rationale for your operation of squaring. You have a sum, represent it X + Y . You are saying if you square a bit of it -> X + Y2 that will be the same thing?

You have something of form A/B2 + C/B .

In general how do you combine that into one?
 

Similar threads

Verify Green's Theorem in the given problem
  • · Replies 10 ·
Replies
10
Views
2K
Question on Two Differentiation Techniques
  • · Replies 25 ·
Replies
25
Views
2K
Solve the given problem involving integration
  • · Replies 6 ·
Replies
6
Views
2K
Calculating Area Under a Curve Using Change of Variables and Quotient Rule
  • · Replies 5 ·
Replies
5
Views
2K
Applying integration to math problems
  • · Replies 5 ·
Replies
5
Views
2K
Differentiate ##f(x)=x\cos{x}+2\tan{x}: D/dx ##\tiny{2.4.2}##
  • · Replies 10 ·
Replies
10
Views
2K
Derivative of two polynomials, one of them being squared
  • · Replies 18 ·
Replies
18
Views
2K
Unsolvable max/min of surface with constraint
  • · Replies 9 ·
Replies
9
Views
2K
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
  • · Replies 2 ·
Replies
2
Views
2K
Dx/x of quotient by def of derivative
  • · Replies 7 ·
Replies
7
Views
1K