Partial derivative of a square root

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Discussion Overview

The discussion revolves around calculating the partial derivative of the equation for the period of a pendulum, specifically the expression 2π√(L/g), with respect to the variable L. Participants are exploring the application of the chain rule and the implications of treating g as a constant in the context of error propagation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about whether they are correctly applying the partial derivative to the expression 2π√(L/g) and mentions using the chain rule.
  • Another participant points out that the original poster has not provided a complete equation, only an expression, which raises questions about the context of the derivative.
  • A participant clarifies that T can be defined as 2π√(L/g), indicating that it is a function of L.
  • There is a suggestion to rewrite the expression to clarify the differentiation process, emphasizing that g is treated as a constant.
  • One participant attempts to apply the chain rule but acknowledges a potential misunderstanding regarding the treatment of g as a constant.
  • Another participant corrects the notation from dL/dT to dT/dL and suggests factoring g out of the square root before differentiation to simplify the process.
  • Concerns are raised about the application of the chain rule versus the power rule in the differentiation process.

Areas of Agreement / Disagreement

Participants express differing views on the correct application of differentiation techniques and the treatment of constants in the expression. There is no consensus on the correct approach to take, and the discussion remains unresolved.

Contextual Notes

Participants note the importance of defining variables clearly and the implications of treating certain variables as constants. There are unresolved questions about the proper application of differentiation rules in this context.

peesha
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Hi,

I'm using partial derivatives to calculate propagation of error. However, a bit rusty on my calculus.

I'm trying to figure out the partial derivative with respect to L of the equation:

2pi*sqrt(L/g)

(Yep, period of a pendulum). "g" is assumed to have no error. I know I can use the chain rule...

so, 2pi*(L/g)^(1/2) --> 2pi*1/2*(L/g)^(-1/2) , or pi*(L/g)^(-1/2).

I am doing this correctly? Or did I just take the derivative (and not the partial derivative)?

Thanks!
 
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peesha said:
Hi,

I'm using partial derivatives to calculate propagation of error. However, a bit rusty on my calculus.

I'm trying to figure out the partial derivative with respect to L of the equation:

2pi*sqrt(L/g)

(Yep, period of a pendulum). "g" is assumed to have no error. I know I can use the chain rule...

so, 2pi*(L/g)^(1/2) --> 2pi*1/2*(L/g)^(-1/2) , or pi*(L/g)^(-1/2).

I am doing this correctly? Or did I just take the derivative (and not the partial derivative)?

Thanks!
Partial derivative of what w.r.t. L? You haven't provided an equation, just an expression.
 
2pi*sqrt(L/g) = T, which is a function of L.
 
Rewrite as T = 2π*(L/g)1/2. Does that give you any ideas? Remember, g is a constant.
 
Using the chain rule, I can bring down the 1/2 and subtract 1 from the exponent, so

dL/dT = 1/2*2π*(L/g)-1/2 or dL/dT = π*(L/g)-1/2

Though, now it seems that I'm not treating "g" as a constant.
 
peesha said:
Using the chain rule, I can bring down the 1/2 and subtract 1 from the exponent, so

dL/dT = 1/2*2π*(L/g)-1/2 or dL/dT = π*(L/g)-1/2

Though, now it seems that I'm not treating "g" as a constant.

Well, factor g out of the square root before taking the derivative.

Technically, you are not using the chain rule. You are using the power rule.
 
peesha said:
Using the chain rule, I can bring down the 1/2 and subtract 1 from the exponent, so

dL/dT = 1/2*2π*(L/g)-1/2 or dL/dT = π*(L/g)-1/2

Though, now it seems that I'm not treating "g" as a constant.
Don't you mean dT/dL? As SteamKing said, you can rewrite the original expression with g outside the square root before you differentiate it. If you don't, you will have to use the chain rule. It tells us that there's an extra factor that you didn't include in the quote above.
 

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