das Produkt
product
An item or service that is made or offered in order to be sold.
Apple hat ein neues Produkt auf den Markt gebracht.
Apple has launched a new product on the market.

product
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product
An item or service that is made or offered in order to be sold.
Apple hat ein neues Produkt auf den Markt gebracht.
Apple has launched a new product on the market.

In an economic context, 'Produkt' (product) is often combined with certain verbs. Typical phrases are 'ein Produkt auf den Markt bringen' (to launch a product on the market), 'ein Produkt entwickeln' (to develop a product), or 'ein Produkt bewerben' (to advertise a product).
Die Firma bringt nächsten Monat ein neues Produkt auf den Markt.
The company is launching a new product on the market next month.
In this meaning, 'Produkt' (product) is often the result of something abstract. You can recognize it in phrases like 'das Produkt von ...' (the product of ...) or 'das Produkt aus ...' (the product of ...), followed by nouns such as 'Arbeit' (work), 'Zufall' (chance), or 'Fantasie' (imagination).
Diese Erfindung ist das Produkt jahrelanger Forschung.
This invention is the product of years of research.
When 'Produkt' (product) describes the result of a process, it is almost always used with the preposition 'von' (of/from) or 'aus' (from/out of) and the dative.
Seine Kreativität ist das Produkt aus Talent und viel Übung.
His creativity is the product of talent and a lot of practice.
You can recognize the mathematical meaning from the context. It is always used in connection with numbers and arithmetic operations such as multiplication.
Was ist das Produkt aus 8 und 12?
What is the product of 8 and 12?
The meaning 'result of a multiplication' belongs to the technical language of mathematics and is not used in normal everyday speech.
Im Mathematikunterricht lernen wir heute, wie man das Produkt zweier Brüche berechnet.
In math class today, we are learning how to calculate the product of two fractions.
article
'der Artikel' and 'das Produkt' can both mean something that is sold. 'Produkt' is the more general term for something that is made or offered, including a service. 'Artikel' more often means an individual item in a shop or warehouse, often as a countable sales unit. In an online shop, one product can have several item variants, for example different sizes or colors. The words are easy to confuse because you talk about both when shopping. In practical terms: 'Produkt' sounds more general, while 'Artikel' is more strongly connected to a product range, an order, or a sales list.

Dieses Produkt ist neu und wird heute zum ersten Mal verkauft.
This product is new and is being sold for the first time today.
Im Lager fehlt noch ein Artikel aus der Bestellung.
One item from the order is still missing in the warehouse.