Derivative of Function f with Respect to t

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In summary, the conversation discusses the possibility of defining a derivative for a function "f" that is a function of "T" but "T" is a function of small "t". The conversation also mentions the chain rule and the use of the Fréchet derivative. The individual asking the question encountered this problem in perturbation analysis and is seeking a reference for further understanding.
  • #1
saravanan13
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I have a function "f", which is a function of "T" but "T" is a function of small "t".
Now my question is what is the derivative of "f" with respect to "t"?
 
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  • #2


What you're asking simply has no sense. Where did you encounter this?

Basically, T could be a function [tex]T:\mathbb{R}\rightarrow \mathbb{R}[/tex] and [tex]f:\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}:T\rightarrow f(T)[/tex].

But now there are two problems
1) I have no clue how to define a derivative on [tex]\mathcal{C}(\mathbb{R},\mathbb{R})[/tex], I'm certain it can be done, but it's not immediately clear.
2) f is not a function of t. The best thing you can do is to define a derivative of f w.r.t. T.

However, you possible can do the following:
define the function [tex]g:\mathbb{R}\times\mathcal{C}(\mathbb{R},\mathbb{R}):(t,T)\rightarrow T(t)[/tex]
And you could possible use this to define a derivative w.r.t. t. But I'm quite sure this is not what you mean...


Where did you encounter this, can you give me the reference??
 
  • #3


I think saravanan13 is talking about the "chain rule":
if y= f(T) is a function to the variable T and T itself is a function of the variable t, then we can think of y as a function of t: y= f(T(t)).

Further, if both functions are differentiable then so is the composite function and
[tex]\frac{dy}{dt}= \frac{df}{dT}\frac{dT}{dt}[/tex]

So that, for example, if [itex]y= T^3[/itex] and [itex]T= 3t^2+ 1[/itex] then we can calculate that [itex]y= (3t^2+ 1)^3= 27t^6+ 27t^4+ 9t^2+ 1[itex] so that
[tex]\frac{dy}{dt}= 182t^5+ 108t^3+ 18t[/tex]

Or we could calculate that
[tex]\frac{dy}{dT}= 3T^2[/tex]
and
[tex]\frac{dT}{dt}= 6t[/tex]
so that
[tex]\frac{dy}{dt}= 3(3t^2+ 1)^2(6t)= 18t(9t^4+ 6t^2+1)= 162t^5+ 108t^2 18t[/tex]
as before.
 
  • #4


micromass said:
What you're asking simply has no sense. Where did you encounter this?

Basically, T could be a function [tex]T:\mathbb{R}\rightarrow \mathbb{R}[/tex] and [tex]f:\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}:T\rightarrow f(T)[/tex].

But now there are two problems
1) I have no clue how to define a derivative on [tex]\mathcal{C}(\mathbb{R},\mathbb{R})[/tex], I'm certain it can be done, but it's not immediately clear.
2) f is not a function of t. The best thing you can do is to define a derivative of f w.r.t. T.

However, you possible can do the following:
define the function [tex]g:\mathbb{R}\times\mathcal{C}(\mathbb{R},\mathbb{R}):(t,T)\rightarrow T(t)[/tex]
And you could possible use this to define a derivative w.r.t. t. But I'm quite sure this is not what you mean...


Where did you encounter this, can you give me the reference??


I came across this problem in perturbation analysis formulated by Ablowitz and Kodama.
In that T is slowly varying time and t is a fast variable.
Thanks for your kin reply...
 
  • #5


micromass said:
What you're asking simply has no sense. Where did you encounter this?

Basically, T could be a function [tex]T:\mathbb{R}\rightarrow \mathbb{R}[/tex] and [tex]f:\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}:T\rightarrow f(T)[/tex].

But now there are two problems
1) I have no clue how to define a derivative on [tex]\mathcal{C}(\mathbb{R},\mathbb{R})[/tex], I'm certain it can be done, but it's not immediately clear.
2) f is not a function of t. The best thing you can do is to define a derivative of f w.r.t. T.

However, you possible can do the following:
define the function [tex]g:\mathbb{R}\times\mathcal{C}(\mathbb{R},\mathbb{R}):(t,T)\rightarrow T(t)[/tex]
And you could possible use this to define a derivative w.r.t. t. But I'm quite sure this is not what you mean...


Where did you encounter this, can you give me the reference??

Could you help me out how to type the mathematics formula in this forum.
After i used some latex that give in the last icon of top left go for a preview it was not shown that i typed.
 
  • #6


micromass said:
I have no clue how to define a derivative on [tex]\mathcal{C}(\mathbb{R},\mathbb{R})[/tex], I'm certain it can be done, but it's not immediately clear.

Just use the Fréchet derivative.
 
  • #7


saravanan13 said:
Could you help me out how to type the mathematics formula in this forum.
After i used some latex that give in the last icon of top left go for a preview it was not shown that i typed.

After you click 'preview', refresh the page - it should now show you what you typed. This is a known issue on these forums.
 

1. What is the definition of a derivative?

The derivative of a function f with respect to a variable t is the rate of change of f with respect to t. It represents the instantaneous slope of the tangent line to the graph of f at a given point.

2. How is the derivative of a function calculated?

The derivative of a function can be calculated using the limit definition of the derivative, which involves taking the limit of the difference quotient as the change in t approaches 0. Alternatively, the derivative can also be found using differentiation rules and formulas.

3. What is the chain rule in calculus?

The chain rule is a rule in calculus that allows us to find the derivative of a composite function. It states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f evaluated at the inner function g(x), multiplied by the derivative of the inner function g'(x).

4. How is the derivative of a function used in real-life applications?

The derivative of a function is used in many real-life applications, such as in physics to calculate velocity and acceleration, in economics to find marginal cost and marginal revenue, and in engineering to optimize designs and systems. It is also used in machine learning and data analysis to find patterns and trends in data.

5. Is the derivative of a function always defined?

No, the derivative of a function is not always defined. It may not exist at points where the function is discontinuous or has a sharp corner, or when the function has a vertical tangent line. The derivative may also be undefined if the function is not differentiable at a certain point.

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