Learn Spanish (Latin America): La prima del doctor es inteligente.

In Spanish, there is no apostrophe 's' for possession. Instead, you swap the order and use de (of). "The doctor's cousin" is literally phrased as "the cousin of the doctor" (la prima de...).

Why is the word del used here instead of just de?

del is a special combination of two words. When the preposition de (of) is immediately followed by the masculine word for "the" (el), they must combine into del. So, de + el doctor becomes del doctor.

How do we know from this sentence that the cousin is a woman?

We can tell because it uses la prima. The "a" at the end of prima and the feminine word for "the" (la) show it is a female cousin. A male cousin would be el primo.

prima is feminine, so why doesn't inteligente end in an "a"?

Adjectives that end in an "e" are gender-neutral in Spanish. inteligente stays exactly the same whether you are describing a man or a woman, so it does not need to change to match la prima.