File: Mathlib/Geometry/Euclidean/Inversion/Calculus.lean

full_name: ContDiffWithinAt.inversion
code: protected theorem ContDiffWithinAt.inversion (hc : ContDiffWithinAt ℝ n c s a)
    (hR : ContDiffWithinAt ℝ n R s a) (hx : ContDiffWithinAt ℝ n x s a) (hne : x a ≠ c a) :
    ContDiffWithinAt ℝ n (fun a ↦ inversion (c a) (R a) (x a)) s a 
goal: ContDiffWithinAt ℝ n (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s a

full_name: ContDiffOn.inversion
code: protected theorem ContDiffOn.inversion (hc : ContDiffOn ℝ n c s) (hR : ContDiffOn ℝ n R s)
    (hx : ContDiffOn ℝ n x s) (hne : ∀ a ∈ s, x a ≠ c a) :
    ContDiffOn ℝ n (fun a ↦ inversion (c a) (R a) (x a)) s 
goal: ContDiffOn ℝ n (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s

full_name: ContDiffAt.inversion
code: protected nonrec theorem ContDiffAt.inversion (hc : ContDiffAt ℝ n c a) (hR : ContDiffAt ℝ n R a)
    (hx : ContDiffAt ℝ n x a) (hne : x a ≠ c a) :
    ContDiffAt ℝ n (fun a ↦ inversion (c a) (R a) (x a)) a 
goal: ContDiffAt ℝ n (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) a

full_name: ContDiff.inversion
code: protected nonrec theorem ContDiff.inversion (hc : ContDiff ℝ n c) (hR : ContDiff ℝ n R)
    (hx : ContDiff ℝ n x) (hne : ∀ a, x a ≠ c a) :
    ContDiff ℝ n (fun a ↦ inversion (c a) (R a) (x a)) 
goal: ContDiff ℝ n fun a => EuclideanGeometry.inversion (c a) (R a) (x a)

full_name: DifferentiableWithinAt.inversion
code: protected theorem DifferentiableWithinAt.inversion (hc : DifferentiableWithinAt ℝ c s a)
    (hR : DifferentiableWithinAt ℝ R s a) (hx : DifferentiableWithinAt ℝ x s a) (hne : x a ≠ c a) :
    DifferentiableWithinAt ℝ (fun a ↦ inversion (c a) (R a) (x a)) s a 
goal: DifferentiableWithinAt ℝ (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s a

full_name: DifferentiableOn.inversion
code: protected theorem DifferentiableOn.inversion (hc : DifferentiableOn ℝ c s)
    (hR : DifferentiableOn ℝ R s) (hx : DifferentiableOn ℝ x s) (hne : ∀ a ∈ s, x a ≠ c a) :
    DifferentiableOn ℝ (fun a ↦ inversion (c a) (R a) (x a)) s 
goal: DifferentiableOn ℝ (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s

full_name: DifferentiableAt.inversion
code: protected theorem DifferentiableAt.inversion (hc : DifferentiableAt ℝ c a)
    (hR : DifferentiableAt ℝ R a) (hx : DifferentiableAt ℝ x a) (hne : x a ≠ c a) :
    DifferentiableAt ℝ (fun a ↦ inversion (c a) (R a) (x a)) a 
goal: DifferentiableAt ℝ (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) a

full_name: Differentiable.inversion
code: protected theorem Differentiable.inversion (hc : Differentiable ℝ c)
    (hR : Differentiable ℝ R) (hx : Differentiable ℝ x) (hne : ∀ a, x a ≠ c a) :
    Differentiable ℝ (fun a ↦ inversion (c a) (R a) (x a)) 
goal: Differentiable ℝ fun a => EuclideanGeometry.inversion (c a) (R a) (x a)

full_name: EuclideanGeometry.hasFDerivAt_inversion
code: /-- Formula for the Fréchet derivative of `EuclideanGeometry.inversion c R`. -/
theorem hasFDerivAt_inversion (hx : x ≠ c) :
    HasFDerivAt (inversion c R)
      ((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x 
goal: HasFDerivAt (EuclideanGeometry.inversion c R)
  ((R / dist x c) ^ 2 •
    ↑{ toLinearEquiv := (reflection (Submodule.span ℝ {x - c})ᗮ).toLinearEquiv, continuous_toFun := ⋯,
        continuous_invFun := ⋯ })
  x