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Braided differential operators $\del^i$ are obtained by differentiating theaddition law on the braided covector spaces introduced previously (such as thebraided addition law on the quantum plane). These are affiliated to aYang-Baxter matrix $R$. The quantum eigenfunctions $\exp_R(\vecx|\vecv)$ of the$\del^i$ (braided-plane waves) are introduced in the free case where theposition components $x_i$ are totally non-commuting. We prove a braided$R$-binomial theorem and a braided-Taylors theorem$\exp_R(\veca|\del)f(\vecx)=f(\veca+\vecx)$. These various results preciselygeneralise to a generic $R$-matrix (and hence to $n$-dimensions) the well-knownproperties of the usual 1-dimensional $q$-differential and $q$-exponential. Asa related application, we show that the q-Heisenberg algebra $px-qxp=1$ is abraided semidirect product $\C[x]\cocross \C[p]$ of the braided line acting onitself (a braided Weyl algebra). Similarly for its generalization to anarbitrary $R$-matrix.Comment: 19 page

Free braided differential calculus, braided binomial theorem, and the braided exponential map