Der Filtereinsatz ist schmutzig.

Because Filtereinsatz is a masculine noun in German.

In the nominative singular (the form used for the subject of the sentence), masculine nouns usually take der:

• der Filtereinsatz
• die Lampe
• das Ventil

In this sentence, der Filtereinsatz is the subject, so der is the correct article.

A useful pattern to remember:

• der = masculine
• die = feminine
• das = neuter

German noun gender usually has to be learned together with the noun.

Yes. Filtereinsatz is a compound noun, which is very common in German.

It is made up of:

• Filter = filter
• Einsatz = insert, unit, cartridge, component placed into something

So Filtereinsatz is literally something like filter insert or filter cartridge, depending on the context.

A very important point about German compounds: the last part usually determines the grammatical gender and the basic category of the word.

Here, the last part is Einsatz, which is masculine, so the whole word is masculine:

• der Einsatz
• der Filtereinsatz

Because German main clauses normally follow the verb-second rule, often called V2.

That means the finite verb comes in the second position in a normal statement.

In this sentence:

• Der Filtereinsatz = position 1
• ist = position 2
• schmutzig = rest of the sentence

So the structure is:

• Subject + verb + complement

This is very typical German word order.

Because schmutzig is being used as a predicate adjective, not as an adjective directly before a noun.

After verbs like:

• sein = to be
• werden = to become
• bleiben = to remain

the adjective usually has no ending.

So:

• Der Filtereinsatz ist schmutzig.
  = schmutzig has no ending

But before a noun, it does take endings:

• der schmutzige Filtereinsatz
• ein schmutziger Filtereinsatz

So compare:

• Der Filtereinsatz ist schmutzig.
• Der schmutzige Filtereinsatz muss ersetzt werden.

It is in the nominative case because it is the subject of the sentence.

You can identify it by asking: What is dirty?

Answer: Der Filtereinsatz.

That makes it the subject, and subjects in German are normally in the nominative.

Here is the masculine singular article pattern for reference:

• der = nominative
• den = accusative
• dem = dative
• des = genitive

So in this sentence, der shows nominative masculine singular.

Because it is the subject complement after sein.

The sentence structure is:

• Der Filtereinsatz = subject
• ist = verb
• schmutzig = adjective describing the subject

This is similar to English:

• The filter insert is dirty.

German often places the complement after the verb in this kind of sentence.

Yes, you probably could, but the tone can be a little different.

Both schmutzig and dreckig can mean dirty, but:

• schmutzig often sounds a bit more neutral or standard
• dreckig can sound a bit stronger, more physical, or slightly more colloquial

In technical or mechanical contexts, schmutzig is often a safe and natural choice.

So:

• Der Filtereinsatz ist schmutzig. = neutral and natural
• Der Filtereinsatz ist dreckig. = also possible, sometimes a bit stronger or less technical

A helpful approximate pronunciation is:

FIL-ter-EIN-zats

Important points:

• Filter sounds roughly like FIL-ter
• ein sounds like English eye-n
• satz has a ts sound at the end, not a simple English z

The main stress is usually on ein in Einsatz:

• FilterEINsatz

And the full sentence is approximately:

der FIL-ter-EIN-zats ist SHMUTS-ikh

A few pronunciation notes:

• sch in schmutzig sounds like English sh
• z in German usually sounds like ts
• the final -ig is often pronounced roughly like -ikh in standard German

In German, all nouns are capitalized, not just proper names.

So Filtereinsatz is capitalized because it is a noun.

Der is capitalized here because it is the first word of the sentence.

Compare:

• Der Filtereinsatz ist schmutzig.
• Ich sehe den Filtereinsatz.

In the second sentence, den is not capitalized because it is not at the beginning.

Yes, but the normal version is:

• Der Filtereinsatz ist schmutzig.

German allows other word orders for emphasis, as long as the finite verb stays in second position in a main clause.

For example:

• Schmutzig ist der Filtereinsatz.

This is grammatical, but it sounds more marked or contrastive, as if you are emphasizing schmutzig.

So for a learner, the safest neutral order is:

• Der Filtereinsatz ist schmutzig.

The plural is usually:

• die Filtereinsätze

Notice two changes:

• the article becomes die
• a in Einsatz changes to ä in the plural: Einsätze

Examples:

• Der Filtereinsatz ist schmutzig.
• Die Filtereinsätze sind schmutzig.

Also notice that the verb changes:

• ist for singular
• sind for plural

Here is the singular pattern for Filtereinsatz:

• Nominative: der Filtereinsatz
• Accusative: den Filtereinsatz
• Dative: dem Filtereinsatz
• Genitive: des Filtereinsatzes

Examples:

• Der Filtereinsatz ist schmutzig.
• Ich reinige den Filtereinsatz.
• Ich arbeite mit dem Filtereinsatz.
• Die Farbe des Filtereinsatzes ist dunkel.

This is useful because masculine nouns change their article depending on case, even though the basic noun often stays similar.