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Simplifying expressions containing Bessel functions?


Is there any kind of Simplification procedure in Mathematica that will 
somehow apply the recursion relations between Bessel functions of orders 
n-1, n and n+1 to eliminate the highest orders in an expression?

I realize this is not a simple topic -- but it would be nice if

   D[x BesselJ[1, x], x]

would yield

   x BesselJ[0, x]

rather than

   BesselJ[1, x]  +  (x/2) ( BesselJ[0, x] - BesselJ[2, x] )


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