Aperture priority mode keeps producing underexposed photos?

Aperture priority mode keeps producing underexposed photos? - 1

Added (1). it's a nikon d40 and for some reason when I try to switch to Auto ISO it tells me it can't be used with the current settings, though I know it can. Trying to shoot outside in natural overcast sunlight so I set the ISO to 400 but no matter whether I try a small or large aperture the photos keep coming out nearly pitch black.

Added (2). in manual mode, i can achieve a proper exposure with a large aperture but when i try to use a small aperture, it still comes out dark though that may be due to my lack of expertise. It's for a class assignment where we're supposed to use both a small and large aperture to compare depth of field using aperture priority mode, and i really can't figure out why it isn't working

Added (3). @efflandt no large aperture works when i'm in manual and adjust everything myself but not when i'm in aperture priority which is what i'm confused about?

Have you tried shooting in manual or adjust your lighting? Can you use Exposure Compensation on your camera?

Are you in an "everything" manual mode? When aperture is manually adjusted, shutter speed needs to be adjusted in the opposite direction (if not automatic). So if a large opening works, and it gets dark when you stop down the lens, the shutter speed needs to be slower. That makes it sound like instead of aperture priority, you might be on something totally manual that is not using automatic exposure or shutter speed.

But you may need to compensate exposure when it is overcast because objects may look darker when back lit by a bright white sky with light from every direction vs. Blue sky with directional light.