How to evaluate all derivatives of a polynomial at a point with FFT?
Hint: Taking the derivative has a certain, simple diagonal effect on the Fourier transform, that hopefully was covered in class. Hence if you compute the Fourier transform of the original function, you "know" the Fourier transform of all its derivatives as well. Given the Fourier transform, there is a formula for the value of the actual function. In your case, in order to compute the value of all derivatives of the function at a point, you multiply some matrix by the vector of Fourier coefficients. This matrix looks suspiciously similar to the DFT matrix, so hopefully you can compute the matrix-vector product in time $O(n\log n)$.