=========
1.

From:
https://en.wikipedia.org/wiki/Recursive_descent_parser

The following EBNF-like grammar for Niklaus Wirth's PL/0 programming language:

 program = block "." .
 
 block =
     ["const" ident "=" num {"," ident "=" num} ";"]
     ["var" ident {"," ident} ";"]
     {"procedure" ident ";" block ";"} statement .
 
 statement =
     ident ":=" expression
     | "call" ident
     | "begin" statement {";" statement } "end"
     | "if" condition "then" statement
     | "while" condition "do" statement .
 
 condition =
     "odd" expression
     | expression ("="|"#"|"<"|"<="|">"|">=") expression .
 
 expression = ["+"|"-"] term {("+"|"-") term} .
 
 term = factor {("*"|"/") factor} .
 
 factor =
     ident
     | number
     | "(" expression ")" .
     
Terminals are expressed in quotes. Each nonterminal is defined by a rule in the grammar, except for ident and number, which are assumed to be implicitly defined.


=========
2.
An Extended-BNF grammar for mathematical expression
bnf001.png

From:
Sahereh Hosseinpour, Mir Mohammad Reza Alavi Milani, Hüseyin Pehlivan
A Step-by-Step Solution Methodology for Mathematical Expressions
Symmetry 2018, 10, 285
p. 5
doi:10.3390/sym10070285

Published: 15 July 2018
https://www.mdpi.com/journal/symmetry

=========
3.
A CFG grammar for quadratic equations
bnf002.png

From:
Mir Mohammad Reza Alavi Milani, Sahereh Hosseinpour, Huseyin Pehlivan
Rule-Based Production of Mathematical Expressions
Mathematics 2018, 6, 254
p. 6
doi:10.3390/math6110254

Published: 15 November 2018
https://www.mdpi.com/journal/mathematics

=========
4.
<a href="https://en.wikipedia.org/wiki/Context-free_grammar#Derivations_and_syntax_trees">Derivations and syntax trees</a> - przykład wyprowadzenia. W szczególności zwracamy uwagę na fragment:
<em>
A derivation of a string for a grammar is a sequence of grammar rule applications that transform the start symbol into the string. A derivation proves that the string belongs to the grammar's language.

A derivation is fully determined by giving, for each step:

- the rule applied in that step,
- the occurrence of its left-hand side to which it is applied.

For clarity, the intermediate string is usually given as well.
</em>