`BookProof.ChapterLaplacianProduct.fderiv_radial` {g : ℝ → ℝ} {x : E} (hx : x ≠ 0) (hg : ContDiffAt ℝ 2 g ‖x‖) : fderiv ℝ (fun y : E => g ‖y‖) x = (deriv g ‖x‖ /...

Proved

BookProof.ChapterLaplacianProduct.fderiv_radial

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLaplacianProduct.

BookProof.ChapterLaplacianProduct.fderiv_radial {g : ℝ → ℝ} {x : E} (hx : x ≠ 0) (hg : ContDiffAt ℝ 2 g ‖x‖) : fderiv ℝ (fun y : E => g ‖y‖) x = (deriv g ‖x‖ / ‖x‖) • innerCLM E x

Formalization note: Lean 4 identifier BookProof.ChapterLaplacianProduct.fderiv_radial.

Preamble

-- Generated from ChapterLaplacianProduct.lean — theorem BookProof.ChapterLaplacianProduct.fderiv_radial
import Mathlib
import Definitions.Def_ChapterLaplacianProduct
import Definitions.Def_ChapterRadialLaplacian
open BookProof.ChapterRadialLaplacian
open BookProof.ChapterLaplacianProduct



open Filter Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open scoped InnerProductSpace RealInnerProductSpace

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

Formal statement

theorem BookProof.ChapterLaplacianProduct.fderiv_radial {g : ℝ → ℝ} {x : E} (hx : x ≠ 0) (hg : ContDiffAt ℝ 2 g ‖x‖) :
    fderiv ℝ (fun y : E => g ‖y‖) x = (deriv g ‖x‖ / ‖x‖) • innerCLM E x := by sorry

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