Can a multiclass Artificer use Alchemical Savant on spell learned through the other class(es)?

Question

The description for Alchemical Savant say:

You've developed masterful command of magical chemicals, enhancing the healing and damage you create through them. Whenever you cast a spell using your alchemist’s supplies as the spellcasting focus, you gain a bonus to one roll of the spell. That roll must restore hit points or be a damage roll that deals acid, fire, necrotic, or poison damage, and the bonus equals your Intelligence modifier (minimum of +1).

Now, say I have multiclassed with a Wizard and learn Infestation (a cantrip available to both Artificer and Wizard) and Chromatic Orb (only on the Wizard's spell list) via that class. Both can do poison damage so they qualify on that part.

Can I take advantage of Alchemical Savant with either spell?

Unlike a lot of other class features, it doesn't say, "When you cast an [insert class name] spell...", just cast a spell and use tools as the focus.

Initially I would just say no, but, while not laid out in the PHB, in XGE it says that the Alchemist's Supplies kit contains:

Alchemist’s supplies include two glass beakers, a metal frame to hold a beaker in place over an open flame, a glass stirring rod, a small mortar and pestle, and a pouch of common alchemical ingredients, including salt, powdered iron, and purified water.

And in the PHB, the [Wizard's Arcane] focuses are described as:

An arcane focus is a special item--an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item--designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus.

So it may be possible to have an arcane focus built-in to the character's alchemist’s supplies and do double duty...

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The is a related question about spells with no material components, but this is about spells that have material components, but learned through a different class.

Answer 1

Your alchemists supplies can only cast artificer spells.

The artificer’s spellcasting feature states:

You produce your artificer spell effects through your tools. You must have a spellcasting focus—specifically thieves’ tools or some kind of artisan’s tool—in hand when you cast any spell with this Spellcasting feature (meaning the spell has an ‘M’ component when you cast it).

On the other hand, the Wizard’s spellcasting feature says:

You can use an arcane focus (see the Adventuring Gear section) as a spellcasting focus for your wizard spells.

An arcane focus and alchemist’s supplies are just two different things. You must use your alchemist’s supplies on artificer spells and your arcane focus on wizard spells.

An arcane focus is a specially designed piece of equipment, not a rod for mashing ingredients into a pestle:

An arcane focus is a special item — an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item — designed to channel the power of arcane spells.

There is no way to make Alchemical Savant work on your wizard spells. Obviously, you can ask your DM, but the rules as written and the rules intent is quite clear that you cannot use Alchemical Savant on anything other than your artificer spells. Ruling otherwise is a house rule, which is fine.

Answer 2

Yes, with the Artificer Initiate feat.

Artificer Initiate reads:

You learn one cantrip of your choice from the artificer spell list, and you learn one 1st-level spell of your choice from that list. Intelligence is your spellcasting ability for these spells.

You can cast this feat's 1st-level spell without a spell slot, and you must finish a long rest before you can cast it in this way again. You can also cast the spell using any spell slots you have.

You gain proficiency with one type of artisan's tools of your choice, and you can use that type of tool as a spellcasting focus for any spell you cast that uses Intelligence as its spellcasting ability.

If you choose alchemist's supplies as your tool, you can use them as the focus for your wizard spells, and for racial spells that use intelligence as their spellcasting ability.