`BookProof.ChapterLaplacianProduct.fderiv_fderiv_mul` {f g : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) : fderiv ℝ (fderiv ℝ fun y : E => f y * g y) x = (g

Proved

BookProof.ChapterLaplacianProduct.fderiv_fderiv_mul

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLaplacianProduct.

BookProof.ChapterLaplacianProduct.fderiv_fderiv_mul {f g : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) : fderiv ℝ (fderiv ℝ fun y : E => f y * g y) x = (g x) • fderiv ℝ (fderiv ℝ f) x + (fderiv ℝ g x).smulRight (fderiv ℝ f x) + ((f x) • fderiv ℝ (fderiv ℝ g) x + (fderiv ℝ f x).smulRight (fderiv ℝ g x))

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

Preamble

-- Generated from ChapterLaplacianProduct.lean — theorem BookProof.ChapterLaplacianProduct.fderiv_fderiv_mul
import Definitions.Def_ChapterRadialLaplacian
import Mathlib
import Definitions.Def_ChapterLaplacianProduct
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_fderiv_mul {f g : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x)
    (hg : ContDiffAt ℝ 2 g x) :
    fderiv ℝ (fderiv ℝ fun y : E => f y * g y) x
      = (g x) • fderiv ℝ (fderiv ℝ f) x + (fderiv ℝ g x).smulRight (fderiv ℝ f x)
        + ((f x) • fderiv ℝ (fderiv ℝ g) x + (fderiv ℝ f x).smulRight (fderiv ℝ g x)) := by sorry

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