It is well known that in the case of integer order differentiation the formula $\partial_{x}f(x,u(x))=\partial_{u}f\cdot \partial_{x}u+\partial_{x}f\cdot u$ holds. If we define fractional derivative via Fourier transform F as $D_{x}^{\alpha}u(x)=F^{1}[{(i\xi)^{\alpha}\widehat{u}(\xi)}]$, where ^ denotes the Fourier transform, is there a similarly formula for $D_{x}^{\alpha}f(x,u(x))$?

If you are interested in estimates (rather than pointwise equalities), you might check out this paper of Christ and Weinstein: http://deepblue.lib.umich.edu/bitstream/handle/2027.42/29171/0000217.pdf?sequence=1 See in particular Proposition 3.1 therein. 

