Is this differentiation correct?

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    Differentiation
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Discussion Overview

The discussion revolves around the correctness of a differentiation performed by a participant, specifically in relation to how a variable I changes with respect to another variable S. The participants explore the implications of this differentiation for understanding sensitivity to changes in S, and they engage in a technical examination of the differentiation process and its application to a specific problem.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant seeks confirmation on the correctness of their differentiation, noting that N, U, and D are constants.
  • Multiple participants challenge the initial differentiation, suggesting it is incorrect.
  • There is a discussion on the effect of differentiation on exponents, with participants proposing alternative forms of the differentiation result.
  • A participant expresses confusion regarding how to interpret the differentiation in terms of sensitivity to changes in S.
  • Another participant suggests using differentials to express the relationship between changes in I and S, providing a formula for dI.
  • Concerns are raised about the interpretation of changes in I when S is increased by 1%, leading to a discrepancy between calculated and expected results.
  • Participants provide calculations based on specific values for N, U, S, and D, leading to different conclusions about the change in I.
  • One participant questions the use of percentages in the calculations, suggesting it complicates the understanding of the results.

Areas of Agreement / Disagreement

Participants generally disagree on the correctness of the initial differentiation and the interpretation of results. There are competing views on how to approach the problem of sensitivity and the calculations involved remain unresolved.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the values of S and the interpretation of percentage changes. Some calculations appear to depend on specific interpretations of the differentiation process and the application of differentials.

Who May Find This Useful

This discussion may be useful for individuals interested in differentiation, sensitivity analysis in mathematical modeling, and those seeking to understand the nuances of applying calculus to real-world problems in physics or engineering contexts.

musicgold
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Hi,

Would you be able to tell me if my differentiation in the attached file correct?
Note that N, U and D are constants.

I am trying to understand how I changes with a change in S.

Thanks.
 

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No, it isn't.
 
What happens to the exponent when we differentiate?
 
HallsofIvy said:
No, it isn't.

Are you sure?
 
SteamKing said:
What happens to the exponent when we differentiate?

Oh! How about this?

dI/dS = (- N U/ D2 ) / ( S U/ D + 1)2
 
That looks better.
 
SteamKing said:
That looks better.

Thanks SteamKing. I am still struggling with one question:
The reason behind this calculation was to understand the sensitivity of I to S. I wanted to know how much I changes for a 1% change in in S. It seems the differentiation doesn't answer that.

What should I do to understand the sensitivity?

Thanks.
 
musicgold said:
Oh! How about this?

dI/dS = (- N U/ D2 ) / ( S U/ D + 1)2
It's much simpler to write the equation in this form:
I = N(US + D)-1

Differentiating with respect to S yields
dI/dS = -N * (US + D)-2 * U = ##\frac{-NU}{(US + D)^2}##

Also, to find the sensitivity of I to small changes in S, write the equation above using differentials.


dI = ##\frac{-NU}{(US + D)^2} dS ##

For small changes in S, you can approximate dS by ΔS and dI by ΔI.
 
Mark44 said:
Also, to find the sensitivity of I to small changes in S, write the equation above using differentials.


dI = ##\frac{-NU}{(US + D)^2} dS ##

For small changes in S, you can approximate dS by ΔS and dI by ΔI.

How do I use this equation to find the change in I for a 1% change in S?
 
  • #10
What Mark is saying (I believe) is that if you replace dI with change in I (the answer you want to get) and dS with change in S (which you have; 1% = 0.01), then you're set. The concept is this: dI/dS is the change of dI over dS; the change of output compared to input. So, in dI/dS, set dS to what it is; the change in input, or 0.01. Now multiply both sides by 0.01 to get dI = (rest of equation). There, you've got a ratio! ^_^
 
  • #11
ModestyKing said:
What Mark is saying (I believe) is that if you replace dI with change in I (the answer you want to get) and dS with change in S (which you have; 1% = 0.01), then you're set. The concept is this: dI/dS is the change of dI over dS; the change of output compared to input. So, in dI/dS, set dS to what it is; the change in input, or 0.01. Now multiply both sides by 0.01 to get dI = (rest of equation). There, you've got a ratio! ^_^

I am still a bit confused. Please see the attached file.
N=100, U= 110, S = 9 and D=120

With these values I = 9.01%
Also these values give me dI/dS = -0.009

Now, for dS = 1%, I get dI= -0.01%. However, this result doesn't make any sense.
For example, if I actually increase S by 1% from 9.00 to 9.09, I changes from 9.01% to 8.93%, a decline of -0.08%, much higher than the 0.01% indicated above.

What am I missing?
 

Attachments

  • #12
You didn't show what you did to get your value of dI, so it's hard to say what you're missing. Using the formula I wrote in post #8, I get dI ≈ -8.93 X 10-5.

I used the values you show in the PDF:
N = 100
U =110
S = 9
D = 120
ΔS = .01
So dI ≈ ## \frac{-100(110)}{(9 * 110 + 120)^2} * .01##

Edit: Corrections to the above[/color]
When S increases by 1%, that means that ΔS = .09[/color], not .01.
This results in a value for dI ≈ ## \frac{-100(110)}{(9 * 110 + 120)^2} * .09 ≈ -8.035 X 10^{-4}##
 
Last edited:
  • #13
Mark44 said:
You didn't show what you did to get your value of dI, so it's hard to say what you're missing. Using the formula I wrote in post #8, I get dI ≈ -8.93 X 10-5.

Sorry about that. You are right, that is how I also calculated it dI= -0.0000893
See the attached Excel file.
 

Attachments

Last edited:
  • #14
Why are you converting to percentages? I think they are throwing you off.

If S increases by 1%, ΔS = .09, so the corrected value for ΔI is about -.0008035.

Calculating I directly with S = 9 results in .09009 (approx.)
Calculating I directly with S = 9.09 results in .08929

So I has decreased (making ΔI negative), and by direct calculation we see that ΔI = -.00080. This agrees reasonably well with the value obtained using differentials.

Using the values I showed in post 12, I get dI ≈ -0.0000893

Calculating I directly with S = 9, I get I ≈ .09009
Increasing S by 1% (to 9.09), I get I ≈ .089293
 
  • #15
Got it. Thank you Mark44. With that correction, it is clear to me now.
:thumbs:
 

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