Commutation of differentiation and averaging operations

plasmoid

I've been studying Turbulence, and there's a lot of averaging of differential equations involved. The books I've seen remark offhandedly that differentiation and averaging commute


for eg. < [tex]\frac{df}{dt}[/tex] > = [tex]\frac{d<f>}{dt}[/tex]

Here < > is temporal averaging. If the differentiation is w.r.t. a spatial coordinate it makes sense, but could someone help me with the above equation?