1. The problem statement, all variables and given/known data

Show that, if Poisson brackets (g,h) = 1, then (g[tex]^{n}[/tex],h) = ng[tex]^{n-1}[/tex]
where g = g(p,q) and h = h(p,q)

p and q are canonical coordinates

3. The attempt at a solution
I suppose that this is purely mathematical, but I am still searching for a detailed example in literature.
I also would like to ask - what book/author can you recommend, where alike problem is discussed.

Thank You!
P.S. I tried search function, but found nothing similar.

ng[tex]^{n-1}[/tex]([tex]\frac{\delta g}{\delta q}[/tex][tex]\frac{\delta h}{\delta p}[/tex] - [tex]\frac{\delta g}{\delta p}[/tex][tex]\frac{\delta h}{\delta q}[/tex])
and the part in brackets is = 1 as we know from given Poisson bracket =>

What have you already seen on Poisson brackets? I could give you the answer but it will make more sense to you if you can get the answer from what you have learned.

First of all, have you seen the definition of a PB?

Yes, I have seen the definition of PB, and brief explanation of its properties.
Should I involve partial integration by time?
I will try to work out what (g^2,h) will give.

You are applying the chain rule, so you should have written, for example,
[tex] \frac{\delta g^n}{\delta q} = n g^{n-1} \frac{\delta g}{\delta q} [/tex] (you forgot the delta g/delta q)
and the same for the derivative with respect to p.

ng[tex]^{n-1}[/tex]([tex]\frac{\delta g}{\delta q}[/tex][tex]\frac{\delta h}{\delta p}[/tex] - [tex]\frac{\delta g}{\delta p}[/tex][tex]\frac{\delta h}{\delta q}[/tex])
and the part in brackets is = 1 as we know from given Poisson bracket =>