Yes, Dell's Specifications for your laptop list that battery as compatible. Whenever touching components or working inside a computer, wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. It's a cheap and sensible precaution. Rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard. (Repeatedly touching a metal case part is not sufficient mitigation with modern components.)
[IMAGE]   I have a Lattitude 5590 that originally had a 68wH/7.6V battery. Recently the battery failed and I bought a 51wH/11.4V battery. The website said it is compatible with my computer, but I want to be positive. Can I use it with my computer? 
Latitude 5590 Battery