Mine nye færdigheder i at danne passive sætninger bliver anvendt i denne lektion.

Because færdigheder is plural.

In Danish, the possessive my changes like this:

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

Since færdigheder means skills and is plural, Danish uses mine:

• min bog = my book
• mit hus = my house
• mine færdigheder = my skills

Because the noun is plural.

Danish adjectives usually take -e in plural forms, and also in definite forms. Here, færdigheder is plural, so ny becomes nye:

• en ny færdighed = a new skill
• nye færdigheder = new skills

So mine nye færdigheder means my new skills.

The basic singular form is færdighed = skill.

The plural is færdigheder = skills.

This is a very common Danish plural pattern:

• singular: færdighed
• plural: færdigheder

The ending -er is a normal plural ending for many Danish nouns.

i at danne means something like in forming or in making.

This structure is common in Danish after nouns like færdighed:

• færdighed i at læse = skill in reading
• færdighed i at skrive = skill in writing
• færdighed i at danne passive sætninger = skill in forming passive sentences

Why both words?

• i is the preposition in
• at danne is the infinitive to form

So literally it is close to skills in to form..., but in real English we would say skills in forming... or skills at forming...

This is just how Danish commonly builds the phrase.

Because after i in this structure, Danish uses the infinitive.

So:

• at danne = to form
• danner = forms / is forming
• dannet = formed

After færdigheder i at..., you need the infinitive:

• færdigheder i at tale dansk
• færdigheder i at skrive breve
• færdigheder i at danne passive sætninger

Because sætninger is plural, so the adjective must match that plural form.

The adjective is based on passiv:

• en passiv sætning = a passive sentence
• et passivt verbum = a passive verb
• passive sætninger = passive sentences

So here:

• sætninger is plural
• therefore passiv becomes passive

Both are passive, but they are built differently.

1. bliver anvendt

This is the periphrastic passive:

• bliver = becomes / is
• anvendt = used

So bliver anvendt means is being used or is used.

2. anvendes

This is the s-passive form of the verb:

• anvende = to use
• anvendes = is used

Both can often be translated the same way.

A similar version of the sentence would be:

• Mine nye færdigheder i at danne passive sætninger anvendes i denne lektion.

That is also grammatical.

Very roughly:

• bliver + past participle often feels a bit more explicit or process-focused
• -s passive is often more compact and common in formal written Danish

Yes.

The tense comes from bliver, which is the present tense of blive.

• bliver anvendt = is used / are used
• blev anvendt = was used / were used
• er blevet anvendt = has/have been used

Even though the subject is plural (færdigheder), Danish still uses bliver. Danish verbs do not change form for singular vs. plural subjects.

So:

• Jeg bliver anvendt — not a natural sentence, but grammatically same verb form
• De bliver anvendt
• Mine færdigheder bliver anvendt

The verb form stays the same.

Because Danish verbs do not agree with the subject in number.

In English, we compare:

• it is
• they are

But in Danish, the verb usually stays the same:

• den bliver
• de bliver

So even though færdigheder is plural, the verb is still bliver.

This is one of the simpler parts of Danish grammar for English speakers.

Because denne means this, while den means that/the depending on context.

So:

• denne lektion = this lesson
• den lektion = that lesson

If the intended meaning is this lesson, then denne is the right choice.

Also, lektion is a common gender noun, so the singular demonstrative is denne.

Compare:

• denne bog = this book
• dette hus = this house
• disse bøger = these books

Yes.

Both are correct, but they feel a little different in style:

• i denne lektion = more formal or written
• i den her lektion = more everyday and spoken

So the sentence could also be:

• Mine nye færdigheder i at danne passive sætninger bliver anvendt i den her lektion.

That sounds a bit more conversational.

This is a normal main clause with the subject first:

• Mine nye færdigheder ... = subject
• bliver anvendt = verb phrase
• i denne lektion = adverbial phrase

So the pattern is basically:

Subject + verb + rest

Yes, Danish can move parts around for emphasis. For example:

• I denne lektion bliver mine nye færdigheder i at danne passive sætninger anvendt.

That version puts i denne lektion first. If you do that in Danish, the finite verb must come second, so bliver comes right after the fronted phrase.

That is standard Danish V2 word order.

It is grammatical, but it sounds a bit formal and slightly heavy because of the long noun phrase and the passive construction.

A Dane might often prefer something simpler, depending on context, for example:

• I denne lektion bruger jeg mine nye færdigheder til at danne passive sætninger.
• Mine nye færdigheder til at danne passive sætninger bliver brugt i denne lektion.

A few notes:

• anvende is correct and a bit formal
• bruge is more everyday
• i at danne is correct
• sometimes til at danne may sound more natural depending on what exact meaning you want

So the original sentence is fine, especially as a grammar-example sentence, but in everyday speech it might be phrased more simply.