Min datter bruger den blå farveblyant til sin tegning.

Because datter is a common-gender singular noun in Danish: en datter.

Danish possessives agree with the gender and number of the thing possessed:

• min = for a singular common-gender noun
• mit = for a singular neuter noun
• mine = for a plural noun

So:

• min datter = my daughter
• mit barn = my child
• mine børn = my children

In Danish, a possessive usually replaces the article.

So you say:

• min datter = my daughter
  not
• en min datter

This works like English, where we say my daughter, not a my daughter.

This is the normal way to make a noun phrase definite when there is an adjective in front of the noun.

Compare:

• en blå farveblyant = a blue colored pencil
• den blå farveblyant = the blue colored pencil

Why den?

Because farveblyant is a common-gender noun: en farveblyant.
So the definite phrase with an adjective uses den:

• den blå farveblyant

If it were a neuter noun, you would use det instead.

Also note that in this structure, Danish uses:

1. a definite article before the adjective (den)
2. the adjective
3. the noun in its base form

So not den blå farveblyanten here.

Adjectives in Danish often change form depending on gender, number, and definiteness, but blå is one of the adjectives whose form is often the same here.

In a definite phrase like den blå farveblyant, the adjective appears in its definite/plural form, which for blå is still blå.

Compare with another adjective where the form changes more clearly:

• en rød blyant = a red pencil
• den røde blyant = the red pencil

With blå, the visible change is not really noticeable in the same way:

• en blå blyant
• den blå blyant

Farveblyant is a compound noun, which is very common in Danish.

It is made from:

• farve = color
• blyant = pencil

So literally it is something like color-pencil, meaning colored pencil or colouring pencil.

Danish often joins nouns together into one word, just like German does and unlike English, which often writes them separately.

Other examples:

• skolebog = schoolbook
• sommerferie = summer holiday
• tandlæge = dentist

So farveblyant is completely normal Danish word formation.

Bruger is the present tense of bruge = to use.

In Danish, verbs do not change according to the person the way they do in English. The same present-tense form is used with all subjects.

Examples:

• jeg bruger = I use
• du bruger = you use
• hun bruger = she uses
• vi bruger = we use

So even though English says she uses, Danish still just uses bruger.

Here til means something like for or for use in connection with.

So til sin tegning means:

• for her drawing
• for use in her drawing

This is a natural Danish way to express purpose.

You will often see til used this way:

• en kop til kaffe = a cup for coffee
• papir til printeren = paper for the printer
• hun bruger rød til blomsten = she uses red for the flower

So in this sentence, til connects the pencil with the drawing it is being used for.

This is a very important Danish grammar point.

Sin/sit/sine is a reflexive possessive, used when the possessor is the subject of the clause.

In this sentence, the subject is:

• Min datter

And the drawing belongs to that same subject. So Danish uses:

• sin tegning = her own drawing

If you said hendes tegning, it would normally suggest the drawing belongs to some other female person, not the daughter herself.

So:

• Min datter bruger den blå farveblyant til sin tegning.
  = My daughter uses the blue colored pencil for her own drawing.

• Min datter bruger den blå farveblyant til hendes tegning.
  = My daughter uses the blue colored pencil for another woman's/girl's drawing.

Because sin/sit/sine must agree with the noun that follows it.

The noun here is tegning, and tegning is a singular common-gender noun:

• en tegning = a drawing

So the correct form is:

• sin tegning

Compare:

• sin bog = his/her own book
• sit hus = his/her own house
• sine bøger = his/her own books

So the choice depends on tegning, not on the person doing the possessing.

Yes. This is very standard Danish main-clause word order:

• Min datter = subject
• bruger = verb
• den blå farveblyant = object
• til sin tegning = prepositional phrase

So the pattern is basically:

Subject + Verb + Object + Prepositional phrase

That is one of the most common and neutral sentence patterns in Danish.

Yes, but it changes the meaning.

• den blå farveblyant = the blue colored pencil
  → a specific one, probably known from context

• en blå farveblyant = a blue colored pencil
  → any blue one, not a specific previously identified pencil

So both are grammatically correct, but they mean different things. The sentence you were given is referring to a specific blue colored pencil.