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**Homework Statement**

If I have the following expansion

[tex]

f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2)

[/tex]

This means for other function U(f(r,t))

[tex]

U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2)

[/tex]

Then up to linear order in ε how to calculate

[tex]

\dfrac{dU}{df} = \ldots O(\varepsilon^2)?

[/tex]

The attempt at a solution

The attempt at a solution

No idea how to approach this:

[tex]

\dfrac{dU}{df} = \dfrac {d U}{d g}\dfrac{dg}{df} = \dfrac{dU/dg}{df/dg} = ?

[/tex]

But then again not sure how to calculate this, if I try

[tex]

\dfrac {d U}{d g} = (U(f(r,t)) - U(g))\dfrac{1}{\varepsilon \delta g}

[/tex]

This will lead me again to dU/dg

Additionally how to calculate dg/df term

EDIT: added details to expression

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