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pixel shader ddx() and ddy() functions

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Hi, I looked up the ddx() and ddy() functions in quite a few places. Everyone says that they compute derivatives. but what exactly do they return? I mean, how do you define 'derivative' at a point in the texture?? also, there is reference to a variant of texNd(sampler, texcoord, ddx, ddy), which is supposed to get the return value from a mipmap depending on the values of ddx and ddy... could anyone elaborate?? Most importantly, is there any proper free HLSL reference on the internet, something like msdn for c++?? the closest I came was 'complete HLSL reference by sebastian laurent' which gives no more than 1 line for each function...

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