composition of functions

[rakuenso]

Hi I have a function f:

 

[math]f(x(t))=\frac{d(x(t))}{dt}+x(t)[/math]

 

Now how would I differentiate f with respect to x

[Royston]

My initial idea was use the chain rule, have you tried solving it with a matrix ?

 

Try separating it with a matrix, you should end up with two differential equations.

[Bignose]

Assuming all the function are well-behaved (i.e. all the derivatives exist) you just take the derivative with respect to x straight forwardly.

 

I'm going to drop the (t) notation -- it is still in there, x is still a function of t, I'm just not going to write it all out.

 

[math]\frac{d}{dx} f = \frac{d}{dx} \left(\frac{dx}{dt} +x\right)[/math]

[math]\frac{d}{dx} f = \frac{d}{dx}\frac{dx}{dt}+\frac{d}{dx}x[/math]

 

now, this is where the "well-behaved" comes in, the order of the differentiation with respect to x & t can be reversed:

 

[math]\frac{d}{dx} f = \frac{d}{dt}\frac{dx}{dx}+\frac{d}{dx}x[/math]

[math]\frac{d}{dx} f = \frac{d}{dt}1+1[/math]

[math]\frac{d}{dx} f = 1[/math]

 

I have no idea what Snail is talking about with matrices and the like. What matrix? where? Furthermore, what is there to "solve"? I really don't get any of Snail's reply.

[Royston]

I was replying to a completely different forum and question, not sure how that happened....sorry about that.

[Bignose]

I was replying to a completely different forum and question, not sure how that happened....sorry about that.

 

That makes a lot more sense -- I can move posts around the math section here, but I don't think they've given me enough power to move posts across forums (...yet).

[rakuenso]

Bignose, under what conditions is

 

[math]\frac{d}{dt}\frac{dx}{dx}=\frac{d}{dx}\frac{dx}{dt}[/math]?

 

I tried to do some examples, say:

 

[math] x = t^2 [/math]

 

[math] \frac{dx}{dt} = 2t [/math] [math]\Rightarrow[/math] [math] \frac{d}{dx}\frac{dx}{dt} = \frac{1}{\sqrt{x}}[/math] since [math] \sqrt{x}=t [/math]

 

However,

 

[math] \frac{d}{dt}\frac{dx}{dx} = 0 [/math]

 

So [math] \frac{d}{dt}\frac{dx}{dx}\neq \frac{d}{dx}\frac{dx}{dt} [/math]

[Bignose]

rak,

 

I'm not ignoring your question, I just haven't had a lot of time to get back to it. My first instinct is that because your f(x) actually isn't a function (which maps a single number to a single number) but is actually a functional (which maps a function to a function), that the definition of differentiation of the function isn't the quite right.

 

see http://mathworld.wolfram.com/FunctionalDerivative.html

 

I need some more time to look at this, so I'll get back when I've thought about it more...