[The equation in the statement of the theorem should read , with rather than as the exponent on the right-hand side.]

I'm assuming you know how to do this in the case n=1. (For a function f(x) of one variable, differentiating the Fourier transform of f corresponds to multiplying f by -2πix, provided that xf(x) is integrable. This is proved by integration by parts.)

To prove the multi-variable case by induction, the inductive hypothesis will be that the result is true when for some integer m>0. The inductive step goes like this. Let be a multi-index with . Then at least one component of , say , must have . Let . Then , so by the inductive hypothesis . To complete the inductive step, all you have to do now is to apply the single-variable result to the function considered as a function of its j'th variable.