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Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts byderivations on a finite dimensional associative or Lie algebra $A$ over a fieldof characteristic $0$. We prove the $\mathfrak g$-invariant analogs ofWedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjectureon asymptotic behavior for codimensions of polynomial identities withderivations of $A$. It turns out that for associative algebras the differentialPI-exponent coincides with the ordinary one. Also we prove the analog ofAmitsur's conjecture for finite dimensional associative algebras with an actionof a reductive affine algebraic group by automorphisms and anti-automorphismsor graded by an arbitrary Abelian group. In addition, we provide criteria for$G$-, $H$- and graded simplicity in terms of codimensions.

Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities