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style(RingTheory/PowerSeries/Derivative): prove derivative_pow from Derivation.leibniz_pow #35067

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style(RingTheory/PowerSeries/Derivative): prove derivative_pow from Derivation.leibniz_pow #35067

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Prove PowerSeries.derivative_pow from Derivation.leibniz_pow
bustercopley Feb 9, 2026
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Merge branch 'master' into derivative_pow
bustercopley Feb 10, 2026
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14 changes: 4 additions & 10 deletions Mathlib/RingTheory/PowerSeries/Derivative.lean
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Expand Up @@ -168,18 +168,12 @@ there is currently no instance of `Inv R⟦X⟧` for more general base rings `R`
apply Derivation.leibniz_of_mul_eq_one
exact PowerSeries.inv_mul_cancel (h := h)

variable (A : Type*) [CommRing A]

/-- The derivative of g^n equals n * g^(n-1) * g'. -/
theorem derivative_pow (g : A⟦X⟧) (n : ℕ) :
theorem derivative_pow (A : Type*) [CommSemiring A] (g : A⟦X⟧) (n : ℕ) :
d⁄dX A (g ^ n) = n * g ^ (n - 1) * d⁄dX A g := by
induction n with
| zero => simp
| succ n ih =>
simp only [pow_succ, (derivative A).leibniz, ih, smul_eq_mul, add_tsub_cancel_right]
rcases n with _ | m
· simp
· simp only [add_tsub_cancel_right, pow_succ]; push_cast; ring
rw [Derivation.leibniz_pow, smul_eq_mul, nsmul_eq_mul, mul_assoc]

variable (A : Type*) [CommRing A]

/-- Chain rule for polynomials viewed as power series. Use `derivative_subst` instead. -/
private theorem derivative_subst_coe (p : Polynomial A) {g : A⟦X⟧} (hg : HasSubst g) :
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