# composition of functions

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Hi I have a function f:

$f(x(t))=\frac{d(x(t))}{dt}+x(t)$

Now how would I differentiate f with respect to x

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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.

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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.

$\frac{d}{dx} f = \frac{d}{dx} \left(\frac{dx}{dt} +x\right)$

$\frac{d}{dx} f = \frac{d}{dx}\frac{dx}{dt}+\frac{d}{dx}x$

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

$\frac{d}{dx} f = \frac{d}{dt}\frac{dx}{dx}+\frac{d}{dx}x$

$\frac{d}{dx} f = \frac{d}{dt}1+1$

$\frac{d}{dx} f = 1$

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.

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

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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).

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Bignose, under what conditions is

$\frac{d}{dt}\frac{dx}{dx}=\frac{d}{dx}\frac{dx}{dt}$?

I tried to do some examples, say:

$x = t^2$

$\frac{dx}{dt} = 2t$ $\Rightarrow$ $\frac{d}{dx}\frac{dx}{dt} = \frac{1}{\sqrt{x}}$ since $\sqrt{x}=t$

However,

$\frac{d}{dt}\frac{dx}{dx} = 0$

So $\frac{d}{dt}\frac{dx}{dx}\neq \frac{d}{dx}\frac{dx}{dt}$

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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.

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

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