Can Mid-2010 MacBook Pro 13" drive 2560 x 1440 external display
While you will be able to drive a display at the resolution with your Macbook Pro 13”, don’t expect the performance to be blazingly fast, and there may be limitations on the type of video connection you can use. Read on for more details.

When you drive a display at that resolution (higher than HD that is) with the graphics card you have in your Macbook, you will be pushing it to it’s maximum. This means if you are looking at a display running 2560x1440 to say play games. You won’t see stealer performance. It will be decent, but you’re not going to be pushing anything over about 30FPS.

If on the other had you are looking at doing coding, web surfing, and the occasional graphics work. Then you should be fine. But video playback in full screen will most likely be choppy at best.

Be aware that the ability to support the resolution you are taking about is not solely the responsibility of the computer. If the display in question uses only a digital video signal (no VGA connector on the monitor) then the monitor must also synchronize with your Mac.  I’ve seen this be a problem with some off brand displays, but an Apple Cinema Display, Dell, HP, or other name brand monitor should work fine in most cases.

For optimal performance you will need to use Dual-link DVI out from your computer, and make sure the monitor supports that. Most monitors support DIV, but not all support the Dual-link DVI format. To do this you will need the Mini DisplayPort to Dual-Link DVI Adapter with cost around $99 as of 2015. By the way, this is REQUIRED if you are using a 30” Apple Cinema Display.

I hope this helps.