Transmute Artifact and Mishra's Workshop

[Komatteru]

Transmute Artifact
Sorcery {U}
As an additional cost to play Transmute Artifact, sacrifice an artifact.
Search your library for an artifact card. If that card's converted mana cost is less than or equal to the sacrificed artifact's converted mana cost, put it into play. If it's greater, you may pay the cost difference. If you do, put it into play. If you don't, put it into its owner's graveyard. Then shuffle your library.

Can you use mana from Mishra's Workshop to pay this cost difference, since Transmute Artifact doesn't explicitly say "pay , where is the cost difference"?

[Jebus]

No.

Workshop can only be used to pay for artifact spells.

You are not paying for an artifact spell here.

[Limbo]

The additional mana you need to pay is like mana leak mana. You can't pay that with workshop either. It has nothing to do with the artifacts actual casting cost(putting it on the stack).