Hey, could someone please guide me as to what to do next in this problem. Thanks!

f(x) = xe^{-(x^2)/2}

Determine the intervals where f is increasing and where f is decreasing.

I've worked out the derivative, but am unsure how to proceed.

Update: I've worked out that the derivative is positive for -1 < x < 1 and negative for x > 1 and x < -1 ... but my working seems quite dodgy, just a bit of guidance would be great. Thanks.