`BookProof.ChapterSolidHarmonicTools.fderiv_normSqPow` (m : ℕ) (x : E) : fderiv ℝ (fun y : E => (‖y‖ ^ 2) ^ m) x = (2 * m * (‖x‖ ^ 2) ^ (m - 1)) • innerCLM E x

Proved

BookProof.ChapterSolidHarmonicTools.fderiv_normSqPow

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.fderiv_normSqPow (m : ℕ) (x : E) : fderiv ℝ (fun y : E => (‖y‖ ^ 2) ^ m) x = (2 * m * (‖x‖ ^ 2) ^ (m - 1)) • innerCLM E x

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonicTools.fderiv_normSqPow.

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.fderiv_normSqPow
import Definitions.Def_ChapterLaplacianProduct
import Mathlib
import Definitions.Def_ChapterSolidHarmonicTools
import Definitions.Def_ChapterRadialLaplacian
open BookProof.ChapterRadialLaplacian
open BookProof.ChapterSolidHarmonicTools



open Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open BookProof.ChapterLaplacianProduct
open scoped RealInnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]

Formal statement

theorem BookProof.ChapterSolidHarmonicTools.fderiv_normSqPow (m : ℕ) (x : E) :
    fderiv ℝ (fun y : E => (‖y‖ ^ 2) ^ m) x = (2 * m * (‖x‖ ^ 2) ^ (m - 1)) • innerCLM E x := by sorry

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