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#1 2005-12-03 01:33:01

johny
Member
Registered: 2005-11-19
Posts: 34

Differentiation!!!!!

I am seriously seriously stuck onto this problem!!!! I really need some help, i need to hand the whole thing by monday, the following question is the only part i am badly stuck on, please help!!!! I need to differentiate this:

= Square of Pi^2*r^6 + 4100625 / r    -  Square of Pi^2*r^6 + 36905625 / 3r.

*Note that the r and 3r dividing the square of above expression are not in square, they are seperate and just dividing.*

Please help.

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#2 2005-12-03 08:20:53

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Differentiation!!!!!

(π² * r^6 + 4100625)² / r  -  (π² * r^6 + 36905625)² / 3r

You will need to use the chain and quotient rules. Here are the pieces:
[ (π² * r^6 + 4100625)² ]′ = 2( π² * r^6 + 4100625 ) * 6π²r^5
= 12π²r^5(π² * r^6 + 4100625)

Quotient rule the first bit:
{  r * [ 12π²r^5(π² * r^6 + 4100625) ] - (π² * r^6 + 4100625)²  } /  r²

Now for the second bit:
[ (π² * r^6 + 36905625)² ]′ = 12π²r^5( π² * r^6 + 36905625)

{  3r * 12π²r^5( π² * r^6 + 36905625) - 3(π² * r^6 + 36905625)²  } / 9r²

Put them together:
( {  r * [ 12π²r^5(π² * r^6 + 4100625) ] - (π² * r^6 + 4100625)²  } /  r² )    -    ( {  3r * 12π²r^5( π² * r^6 + 36905625) - 3(π² * r^6 + 36905625)²  } / 9r² )

I'm sure you could simplify that quite a bit, but it's hard to see on a computer screen. smile

Last edited by ryos (2005-12-03 08:42:54)


El que pega primero pega dos veces.

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#3 2005-12-03 08:26:58

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Differentiation!!!!!

I think that you meant:

   y = (pi^2 r^6 + 4100625)^(1/2) / r  - (pi^2 r^6 + 36905625)^(1/2) / 3r

     There are no shortcuts here and the answer will not be pretty.  It is easier if you treat each part of this equation as a separate function and then add them back together after differentiating.
We will call the first part A and the second part B.  Also the constants have no effect upon the outcome so we will call them a and b respectively.

  A = (pi^2 r^6 + a)^(1/2) / r  = [(pi^2 r^6 + a) / r^2)]^(1/2)

  dA/dr = {1/2 [(pi^2 r^6 + a) / r^2]^(-1/2)} {[(6 pi^2 r^5)r^2 - 2r(pi^2 r^6 + a)] / r^4}

  dA/dr = (6 pi^2 r^7 - 2 pi^2 r^7 - 2ar) / [(4 pi^2 r^14 + 4a r^8) / r^2]^(1/2)

  dA/dr = (4 pi^2 r^7 - 2ar) / [4 pi^2 r^12 + 4a r^6]^(1/2)

  da/dr = (4 pi^2 r^7 - 2ar) / 2 r^3 [(pi^2 r^6 + a)^(1/2)]

  da/dr =  (2 pi^2 r^6 - a) / (pi^2 r^10 + a r^4)^(1/2)


  B = (pi^2 r^6 + b)^(1/2) / 3r  =  [(pi^2 r^6 + b) / 9 r^2]^(1/2)

db/dr = {1/2[(pi^2 r^6 + b)/(9 r^2)]^(-1/2)}{[6 pi^2 r^5(9 r^2) - 18r(pi^2 r^6 + b)]/81 r^4}
 
db/dr = (54 pi^2 r^7 - 18 pi^2 r^7 + 18br)/[54 r^3(pi^2 r^6 + b)^(1/2)]

db/dr = (36 pi^2 r^7 + 18br) / [54 r^3(pi^2 r^6 + b)^(1/2)]

db/dr = (2 pi^2 r^6 + b) / [3 r^2 (pi^2 r^6 + b)^(1/2)]

whew....

dy/dr = {(2 pi^2 r^6 - a) / (pi^2 r^10 + a r^4)^(1/2)}
            - (2 pi^2 r^6 + b) / [3 r^2 (pi^2 r^6 + b)^(1/2)]

     This is why you have not gotten any answers.  It is too long a computation just for the sake of argument.  I have just found the rate of change for an unknown function.  It is pretty dissatisfying.

     Perhaps you could tell us what the function represented.

Last edited by irspow (2005-12-03 08:29:31)

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#4 2005-12-03 09:01:35

johny
Member
Registered: 2005-11-19
Posts: 34

Re: Differentiation!!!!!

Thanks ryos and irspow for spending time on my question smile......Irspow, well the formula was originally for the volume and surface area of a cone, my shape was a truncated cone, but as i had too many variables in the formula i.e height, radius. I had to remove one of them and i did it by saying the short part of the cone is 1/3 of the whole cone......what i did next was re-arranged the formula for volume to take height, then i put the equation of height in surface area.

I had volume fixed to 600cm^3.

Then i substituted the equation of height into the surface area. It came to something like this:

S.A= Pi*R * square root of R^2 + 2025/Pi*R^2

Then i had to subtract this equation by 1/3 R and 1/3 of H, when i simplified it came to the equation which i gave u!!!!! big_smile.

Hope u understand. And thanks once again.

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#5 2005-12-04 01:57:37

johny
Member
Registered: 2005-11-19
Posts: 34

Re: Differentiation!!!!!

I didnt want to start a new topic on this but i am very confused about the following:


   = Pi* R * S          where S= Square root of H^2 + R^2

U know i was doing a question for truncated cone.........so what i did was i put some dotted lines on that truncated cone to make it like a cone and i said that the dotted lines were 1/3 of the whole cone, as i wanted to find the surface area, meaning the radius is 1/3, height is 1/3 and slant height (s) is 1/3.  So can we write:

        = Pi * R/3 * Square root of R^2 + H^2  /  3  ????    (note: 3 is not in square root, its seperate)

In simple numbers i have done :  Pi*r/3*s/3

I hope u understand the problem????

Last edited by johny (2005-12-04 01:58:54)

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#6 2005-12-05 05:40:50

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Differentiation!!!!!

This relationship between radius, height, and side(slant height) can not exist, at least not in a cone.  If your height and radius were 1/3 the side would be the square root of 2/9 or ≈.471

    I have tried to follow this thread and your previous posts and have still not figured out exactly what the original problem was.  It seems that you have been going along for a while now on this problem. 

     I think that if you posted a more concise definition of the original problem that you would be pleasantly surprised by a more simple solution.  It sounds like some type of basic related rate problem which ususally can be solved quite simply with all of the relevant information from the beginning.

    Forgive me if I sound like some type of a-hole, but I think that you somehow made this problem more difficult than it truly needs to be.  I, and I suspect others here, would love to help you, but your problem never seemed to be exactly clear from the beginning.

Last edited by irspow (2005-12-05 05:48:36)

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