How to show that given language is unambiguous Given following grammar:

$$
\begin{align}
S \rightarrow &amp;A1B \\
A \rightarrow &amp; 0A \mid \varepsilon \\
B \rightarrow &amp; 0B \mid 1B \mid \varepsilon \\
\end{align}
$$

How can I show that this grammar is unambiguous? I need to find a grammar for the same language that is ambiguous, and demonstrate it.

I know if I was asked to prove that the language is ambigious then I should find two different parse trees for same string, but I don't know what to do.
To show a grammar is unambiguous you have to argue that for each string in the language there is only one derivation tree.

In this particular case you can observe that $A$ only generates $0$'s, so the $1$ generated by the start symbol $S$ must be the first $1$ in the string. 

Any grammar can be made ambiguous by adding chain productions like $S\to S$.