Direct quoting from the DnD Beyond page for Rage, a Barbarian 1st level ability:

If you are able to cast spells, you can’t cast them or concentrate on them while raging.

Direct quote from the DnD Beyond page for Spellsmith, a Variant Human: Mark of Making innate ability:

You know the mending cantrip. You can also cast the magic weapon spell with this trait. When you do so, the spell lasts for 1 hour and doesn’t require concentration. Once you cast the spell with this trait, you can’t do so again until you finish a long rest. Intelligence is your spellcasting ability for these spells.

If I cast magic weapon first with Spellsmith, and then Rage, will it cancel magic weapon even though I don't have to concentrate on it?

The Barbarian 1st level ability Rage states:

If you are able to cast spells, you can’t cast them or concentrate on them while raging.

And the Variant Human: Mark of Making innate ability Spellsmith states:

You know the mending cantrip. You can also cast the magic weapon spell with this trait. When you do so, the spell lasts for 1 hour and doesn’t require concentration. Once you cast the spell with this trait, you can’t do so again until you finish a long rest. Intelligence is your spellcasting ability for these spells.

If I cast magic weapon first with Spellsmith, and then Rage, will it cancel magic weapon even though I don't have to concentrate on it?

Direct quoting from the DnD Beyond page for Rage, a Barbarian 1st level ability:

"If you are able to cast spells, you can’t cast them or concentrate on them while raging."

Direct quote from the DnD Beyond page for Spellsmith, a Mark of Making Human innate ability:

"You know the mending cantrip. You can also cast the magic weapon spell with this trait. When you do so, the spell lasts for 1 hour and doesn’t require concentration. Once you cast the spell with this trait, you can’t do so again until you finish a long rest. Intelligence is your spellcasting ability for these spells."

If I cast Magic Weapon first with Spellsmith, and then Rage, will it cancel Magic Weapon even though I don't have to concentrate on it?

Direct quoting from the DnD Beyond page for Rage, a Barbarian 1st level ability:

If you are able to cast spells, you can’t cast them or concentrate on them while raging.

Direct quote from the DnD Beyond page for Spellsmith, a Variant Human: Mark of Making innate ability:

You know the mending cantrip. You can also cast the magic weapon spell with this trait. When you do so, the spell lasts for 1 hour and doesn’t require concentration. Once you cast the spell with this trait, you can’t do so again until you finish a long rest. Intelligence is your spellcasting ability for these spells.

If I cast magic weapon first with Spellsmith, and then Rage, will it cancel magic weapon even though I don't have to concentrate on it?