I've posted this to Math.SE about a month ago:
Seems like $$ \Delta(a_0+a_1t^d+a_2t^{2d}+...+a_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}[\Delta(a_0+a_1t+a_2t^2+...+a_nt^n)]^d, $$ where $\Delta$ is the discriminant.
Presumably this is not difficult to prove, but I just need a reference. Who did discover this formula for the first time?
You say "presumably not difficult to prove." Did you try? It seems like a pretty easy exercise, using $$ \Delta(F(t)) = \prod_{F(a)=0} F'(a). $$ Taking $F(t)=f(t^d)$, the roots of $F$ are the $d$'th roots of the roots of $f$, while $F'(t)=dt^{d-1}f'(t^d)$. So I doubt you'll find this formula in a reference, but if you need to use it in a paper, you can just state it as a lemma, and for the proof, say "Exercise using the formula $ \Delta(F(t)) = \prod_{F(a)=0} F'(a) $ with $F(t)=f(t^d)$."
I don't know of a standard reference, but there is a note by John Cullinan at Bard College where he computes the discriminant of a composition of two polynomials, so a nontrivially more general result than the one you mention.
