to imitate / copy
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to imitate
To imitate or copy another person's actions, voice, or behavior, often in a playful or mocking way.
Mein kleiner Bruder versucht immer, alles nachzumachen, was ich tue.
My little brother always tries to imitate everything I do.
'Nachmachen' (to imitate / copy) is a separable verb. The prefix 'nach' comes at the end in a main clause, but remains attached to the verb in a subordinate clause. In the infinitive with 'zu', 'zu' goes between 'nach' and 'machen'.
Pass auf, dass die Kinder im Chemieunterricht nichts Gefährliches nachmachen.
Make sure the children don't copy anything dangerous in chemistry class.
The prefix 'nach-' here has the meaning of 'afterward' or 'as the second one'. So 'nachmachen' (to imitate / copy) literally means to do something after someone else has demonstrated it.
Er sang ein Lied und alle machten es ihm nach.
He sang a song and everyone copied him.
When you specify whose action you are imitating or copying, that person or thing is in the dative. The structure is often: 'jemandem (dative) etwas (accusative) nachmachen' (to imitate/copy something from someone).
Meine kleine Schwester macht mir alles nach.
My little sister copies everything I do.
The object of the sentence makes the meaning clear. If it is a person, a gesture, or a sound, 'imitieren' (to imitate) is meant. If it is a product, a recipe, or a document, 'reproduzieren' (to reproduce) or 'fälschen' (to counterfeit) is meant.
Der Komiker kann perfekt die Stimmen von Politikern nachmachen.
The comedian can imitate the voices of politicians perfectly.
To express that an achievement is extraordinary, the negated form is often used: 'Das macht ihm so schnell keiner nach.' (No one will match that anytime soon.) This is an expression of admiration.
Mit 90 Jahren noch einen Marathon laufen? Das macht ihr so schnell keiner nach!
Running a marathon at 90? No one will match her anytime soon!
to keep up
'mithalten' and 'nachmachen' can both have to do with another person's performance. 'mithalten' means that you stay at the same level or manage the same pace. 'nachmachen' here means that you also manage a special achievement. It is often used in sentences like 'Das macht ihm keiner nach' (no one can match him in that). The words can be confused when someone is very good and others try to be just as good. 'mithalten' emphasizes the ongoing comparison, while 'nachmachen' emphasizes repeating or achieving a specific accomplishment.
So ein schwieriges Kunststück macht ihr im Verein fast niemand nach.
Almost no one in your club can match such a difficult trick.
In der schnellen Diskussion kann ich nicht immer mithalten.
In the fast-paced discussion, I can't always keep up.