I am having trouble solving this question. I know you are supposed to use the limit defnition, but I can't figure it out.

Let F: R^n -> R^m be a differentiable mapping. Given u, v are vectors in R^n, show that

D_(au+bv)F = aD_u(F) + bD_v(F), where a, b: Rn -> R are arbitrary functions.