In GPU Pro 2, there's a chapter called "Shader Amortization using Pixel Quad Message Passing", which explains how you can use the derivative functions to communicate between different instances of a pixel shader function, which allows for certain kinds of optimisations to be made.
I've personally used this technique to output to a R10_G10_B10_A2 render target, as if it was a full-resolution R10_G10_B10 target and a half-resolution A8 target, which was pretty useful.
Back to the point, this kind of inter-pixel communication is only possible with "high quality" derivatives. With "low quality" derivatives, only 3 out of every 4 pixels can communicate, with 1 out of 4 pixels being isolated and unable to talk to the others in it's "2x2 quad".
As far as I know, there's no way to query the hardware to see whether "low" or "high" quality derivatives are the default... You can only discover this by attempting to use one of these communication algorithms, and then reading the results to see if it worked or not.
I guess that if you wanted to use one of these techniques, you could first use the GL hint to kindly request "high" quality derivatives, and then test if your algorithm works, to increase your chances.
n.b. as of DX10/GL3 level hardware, all GPUs support both "low" and "high" quality modes (HLSL has different instructions for each, with the old "dFdx" type instructions being a "don't care" option).