For $f=a_0+a_1t+...+a_nt^n$, seems like $$ \Delta(f(t^d))=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}\Delta(f)^d, $$ 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?