Because PhotoRealistic RenderMan is a micropolygonbased renderer, it has always based its derivatives and differentials on the current micropolygon size; that is, if the micropolygon subtends a certain range in u and v, the derivatives will reflect a shading element of exactly that size. This is an intuitive representation of the derivatives, but can sometimes introduce subtle filtering artifacts when adjacent regions of an object have been diced at different rates. We seek a shading element size which is independent of the actual dicing rate of the object.
Recall that PRMan's "shading rate" attribute indicates the desired micropolygon size in screen space; the renderer, when dicing a primitive into micropolygons, will try to achieve this rate, but rarely hits it exactly. Across a shading grid, some micropolygons may project to slightly larger than the "perfect" size, and some may be slightly smaller. As a result, micropolygons on adjacent grids may have significantly different sizes, and this leads to discontinuities in the derivative functions.
PRMan 3.8 uses a new method to compute surface derivatives and differentials which is independent of the actual dicing rate of the object. The renderer now estimates the size of a "perfect" micropolygon (one which projects to exactly the shading rate in size), and bases its derivative values on that. The result is continuous derivative functions, regardless of the actual dicing rate.
If you suspect underfiltering may be causing aliasing on a given object, smooth derivatives can be deactivated on a pershader basis. The shader compiler has two new commandline parameters which turn on and off the use of smooth derivatives:
shader sd <shadername>  [turns smooth derivates on (default)] 
shader nosd <shadername>  [turns smooth derivates off] 
When smooth derivates are turned on, the global variables du and dv represent the smooth differentials, and when smooth derivatives are turned off, du and dv represent the oldstyle, gridbased differentials. Regardless of the compilation mode, you can access the two styles of derivatives explicitly with the following four global variables:
Shaders which are sensitive to different derivative methods can pickandchoose with these variables.
Finally, note that the internal filtering functions (texture(), shadow(), environment(), filterstep(), area()) use smooth derivatives automatically when sd is specified. If the aforementioned aliasing problems affect your use of these builtin functions, simply turn off smooth derivatives with nosd when compiling the shader.
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