急かす【せかす】 is transitive, and usually takes a person as an object.
- 彼を急かした。 I hurried him. / I made him hurry.
- そんなに(俺を)急かすなよ。 Don't rush me so much.
急ぐ【いそぐ】 is usually intransitive, but it also sometimes takes を as a location marker. 急ぐ also has a transitive usage, and when 急ぐ is transitive, it takes an inanimate object.
- 私は今、急いでいます。 I'm in a hurry now.
- この道を急ぎましょう。 Let's hurry in this way.
- プロジェクトの完成を急ごう。 Let's hurry to complete the project.
Etymologically, I believe 急かす【せかす】 is one of the causative forms of 急く【せく】, which is another rare intransitive verb meaning to hurry. It's the same as the "shortened causative form" explained here, but 急かす is far more common in 急く today. 急く is found almost only in this proverb. I think you can regard 急かす as a separate (transitive) verb.