# I have a function

1. ### saravanan13

56
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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3. ### micromass

19,229
Staff Emeritus
Re: derivative

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

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

But now there are two problems
1) I have no clue how to define a derivative on $$\mathcal{C}(\mathbb{R},\mathbb{R})$$, 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 $$g:\mathbb{R}\times\mathcal{C}(\mathbb{R},\mathbb{R}):(t,T)\rightarrow T(t)$$
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??

4. ### HallsofIvy

40,687
Staff Emeritus
Re: derivative

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
$$\frac{dy}{dt}= \frac{df}{dT}\frac{dT}{dt}$$

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

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

5. ### saravanan13

56
Re: derivative

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.

6. ### saravanan13

56
Re: derivative

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.

7. ### pwsnafu

931
Re: derivative

Just use the Fréchet derivative.

8. ### plasmoid

15
Re: derivative

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