Use elementary row operations. If you add a multiple of one row to another row then the determinant is unchanged. So subtract row 1 from row 2 and from row 3, and you get

Now you know that y–x is not zero, so you can divide through row 2 by y–x; and you can divide through row 3 by z–x. That gives you

Continue to simplify the determinant by similar manoeuvres. For example, you could now subtract row 2 from row 3, and subtract x times row 2 from row 1. By then, the determinant should look simple enough that you can work it out completely.