Den her jakke er dyrere end den der.

In Danish, this and that are often built with:

• den/det/de
  • noun + her/der

So:

• den her jakke = this jacket
• den der jakke = that jacket

Literally, den her jakke is something like the here jacket, but in real Danish it simply means this jacket.

This is a very common everyday pattern.

• den is used with common gender nouns
• det is used with neuter nouns
• de is used with plurals

Since jakke is a common gender noun, it takes den.

Because jakke is a common gender noun in Danish.

Danish has two grammatical genders:

• common gender → takes en in the indefinite form and den in this kind of structure
• neuter → takes et in the indefinite form and det

You say:

• en jakke = a jacket
• therefore den her jakke = this jacket

If the noun were neuter, you would use det instead:

• et hus = a house
• det her hus = this house

In this sentence:

• her means here and functions as this
• der means there and functions as that

So:

• den her jakke = this jacket
• den der = that one / that one there

These little words help show distance:

• her = near the speaker
• der = farther away

They are very common in spoken Danish.

Because Danish can leave out the noun when it is already understood from context.

So:

• Den her jakke er dyrere end den der.

literally means:

• This jacket is more expensive than that one.

The word jakke is omitted in the second part because it would be repetitive. English does the same thing with one:

• this jacket ... that one

In Danish, den der by itself can stand in for that jacket when the noun is obvious.

Dyrere is the comparative form of dyr (expensive).

So:

• dyr = expensive
• dyrere = more expensive

This is like English:

• small → smaller
• cheap → cheaper

In the sentence:

• Den her jakke er dyrere ...
• This jacket is more expensive ...

Danish often forms comparatives by adding -ere.

End means than in comparisons.

So:

• dyrere end = more expensive than
• større end = bigger than
• bedre end = better than

In your sentence:

• Den her jakke er dyrere end den der.
• This jacket is more expensive than that one.

So whenever you compare two things, end is usually the word you need.

Yes, you can, but it sounds more formal or written.

There are two common ways to say this in Danish:

1. den her jakke — very common in everyday speech
2. denne jakke — more formal, more written, sometimes more emphatic

So both are correct, but they feel a little different.

• Den her jakke = natural, everyday
• Denne jakke = more formal or stylistically marked

For most casual spoken Danish, den her jakke is the safest choice.

No, den der is not automatically rude in Danish.

Very often it is just a normal way to mean:

• that
• that one
• the one over there

However, tone of voice matters. If someone stresses it in a certain way, it can sound annoyed or dismissive, just as in English. But by itself, in a neutral sentence like this one, it is completely normal.

This sentence is a normal main clause in Danish, so the basic word order is:

• subject + verb + complement

Here that is:

• Den her jakke = subject
• er = verb
• dyrere end den der = complement

So the structure is:

• This jacket
  • is
    • more expensive than that one

If something else were placed first for emphasis, Danish would change the order because of the verb-second rule, but in this simple sentence the word order is straightforward.

Normally, in this pattern, no.

The usual everyday structure is:

• den her jakke
• det her hus
• de her sko

not:

• her jakke
• der jakke

So her and der come after den/det/de and usually before or after the noun depending on the structure being used. In this common spoken pattern, the full phrase is:

• den her jakke

You may also see another structure with denne/dette/disse, where there is no her/der:

• denne jakke = this jacket
• dette hus = this house
• disse sko = these shoes

In the plural, Danish uses de.

So:

• de her jakker = these jackets
• de der = those / those ones

A full sentence would be:

• De her jakker er dyrere end de der.

That means:

• These jackets are more expensive than those.

Notice that:

• singular common gender: den her jakke
• plural: de her jakker

Yes, it is very natural.

Den her jakke er dyrere end den der sounds like normal spoken Danish. It is clear, idiomatic, and common.

Other versions are possible, such as:

• Denne jakke er dyrere end den der.
• Den her jakke er dyrere end den anden. (if you mean the other one rather than that one over there)

But your original sentence is a very natural everyday way to say it.