How do I calculate the differential of f(x+dx)?

[graupner1000]

Hi all,

I'm a bit stuck on what should probably be fairly simple, but I'm looking for a general way to do

$\partial_{x}f(x+\delta x)$

Any help would much appreciated.

 

[mathman]

The notation is a little off beat. I've never seen partial differentials, only partial derivatives.

 

[deluks917]

I'm not quite sure what is meant either. Is δ a number or a function. Is f a function of more than one variable (if not why the partial derivative notation)?

 

[graupner1000]

Sorry, just some old habits. What I mean is this: A function f(x) is perturbed by δx so f(x+δx). What is the differential of this:

$\frac{d}{dx}f(x+\delta x)$

which is what I think it should actually look like.

 

[Stephen Tashi]

A function f(x) is perturbed by δx

If you mean $\delta x$ to be a finite constant then this is like asking "What is the differential of f(x+5)?" Is it that sort of question?

Or are you doing some sort of reasoning involving "infinitesimals"? If so, it would be better to give a complete context for the situation.

 

[graupner1000]

Ok, if I'm going to describe the full problem this should go in the cosmology section.

The goal is to rewrite the Klein-Gordon equation in terms of a perturbation to the scalar field. So, starting from

$\frac{d^{2}\phi}{dt^{2}} + 3H\frac{d\phi}{dt} + \frac{dV}{d\phi}$=0

where $\phi=\phi(x,t)$ and $V=V(\phi)$

and using

$\phi(x,t)=\phi(t)+\delta\phi(x,t)$

I have gotten as far as

$\frac{d^{2}\phi}{dt^{2}} + \frac{d^{2}\delta\phi}{dt^{2}} +3H\frac{d\phi}{dt} + 3H\frac{d\delta\phi}{dt} + \frac{dV(\phi + \delta\phi)}{d\phi} = 0$

Now what I am trying to get is

$\frac{d^{2}\delta\phi}{dt^{2}} + 3H\frac{d\delta\phi}{dt} +\frac{d^{2}V}{d\phi^{2}}\delta\phi =0$

So you see what I meant with my original post. I figured If I could evaluate the last term I'd get the correct answer but I can't remember how to do it.

 

[bigfooted]

you should now subtract the original unperturbed equation and apply the differentiation rule: df/dx = f(x+dx) - f(x) which immediately leads to the result.

 

[graupner1000]

It worked thanks allot. I knew it would be something easy.