La partie la plus difficile du livre est le dernier chapitre.

Because partie is a feminine noun in French. French nouns have grammatical gender, and partie takes the feminine article la.

So:

• la partie = the part
• une partie = a part

This gender is grammatical, not about natural sex.

This is the superlative structure: the most difficult.

It is formed with:

• la = the
• plus = more / most
• difficile = difficult

So la plus difficile means the most difficult.

It agrees with partie, which is feminine singular, so the article is la.

Compare:

• la partie la plus difficile = the most difficult part
• le chapitre le plus difficile = the most difficult chapter

Because difficile already has the same form in the masculine and feminine singular.

Many French adjectives change:

• petit → petite
• grand → grande

But difficile stays:

• difficile (masculine singular)
• difficile (feminine singular)

Only the article shows the gender clearly here: la plus difficile.

The first la is the article for the noun:

• La partie = the part

The second la is part of the superlative phrase modifying the noun:

• la plus difficile = the most difficult

So the whole phrase works like:

• La partie
  • la plus difficile
• The part
  • the most difficult

French often repeats the definite article in this structure.

Because de + le contracts to du in French.

So:

• de le livre → du livre

This is a standard contraction:

• de + le = du
• de + les = des
• à + le = au
• à + les = aux

Here du livre means of the book.

It shows possession or relationship: the part of the book.

So:

• la partie du livre = the part of the book

Then French adds the description:

• la partie la plus difficile du livre = the most difficult part of the book

Est is the third-person singular form of the verb être (to be).

The subject is La partie la plus difficile du livre, which is singular, so French uses est:

• elle est = it is / she is

So:

• La partie ... est ... = The part ... is ...

Because chapitre is a masculine singular noun, so it takes le.

Also, dernier agrees with chapitre:

• le dernier chapitre = the last chapter

If the noun were feminine, it would change:

• la dernière page = the last page

Some French adjectives usually come before the noun, and dernier is one of them in this meaning.

So French says:

• le dernier chapitre

not normally:

• le chapitre dernier

This is one of those adjective-position patterns you gradually learn. Common adjectives that often go before the noun include ones like petit, grand, jeune, vieux, bon, and dernier.

Usually in a sentence like this, yes: the last or final.

So:

• le dernier chapitre = the last chapter / the final chapter

In other contexts, dernier can sometimes mean previous depending on position and meaning, but here it clearly means final.

French uses articles more often than English does, and each one has a job:

• La partie: article for the noun partie
• la plus difficile: article used in the superlative
• du livre: contraction of de + le
• le dernier chapitre: article for chapitre

English often uses fewer visible markers, but French grammar likes to make these relationships explicit.

Yes, in some contexts, but it changes the nuance a bit.

• difficile = difficult
• dur / dure = hard, tough

For a book or chapter, difficile is usually the more neutral and natural choice if you mean intellectually difficult or challenging.

So La partie la plus difficile du livre is a very natural sentence.

Because it is identifying one thing as one thing:

• Subject: La partie la plus difficile du livre = one part
• Complement: le dernier chapitre = one chapter

The sentence is saying that these two refer to the same thing:

• The most difficult part of the book = the last chapter

So singular on both sides is exactly what you expect.

A careful pronunciation would be roughly:

La par-tee la plu dif-fee-seel du leevr ay luh der-nyay sha-pee-truh

A more natural French rhythm is smoother than that, and a few sounds to notice are:

• partie sounds like par-tee
• plus in this sentence is usually pronounced plu
• difficile ends with an l sound: dif-fee-seel
• livre has the French r
• dernier sounds like der-nyay
• chapitre sounds like sha-peetr

There is no major required liaison here that changes the sentence dramatically, but the whole sentence flows together in normal speech.