File: Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean

full_name: Real.deriv_arcsin_aux
code: theorem deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x 
goal: HasStrictDerivAt Real.arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ Real.arcsin x

full_name: Real.hasStrictDerivAt_arcsin
code: theorem hasStrictDerivAt_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x 
goal: HasStrictDerivAt Real.arcsin (1 / √(1 - x ^ 2)) x

full_name: Real.hasDerivAt_arcsin
code: theorem hasDerivAt_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasDerivAt arcsin (1 / √(1 - x ^ 2)) x 
goal: HasDerivAt Real.arcsin (1 / √(1 - x ^ 2)) x

full_name: Real.contDiffAt_arcsin
code: theorem contDiffAt_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞} : ContDiffAt ℝ n arcsin x 
goal: ContDiffAt ℝ n Real.arcsin x

full_name: Real.hasDerivWithinAt_arcsin_Ici
code: theorem hasDerivWithinAt_arcsin_Ici {x : ℝ} (h : x ≠ -1) :
    HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Ici x) x 
goal: HasDerivWithinAt Real.arcsin (1 / √(1 - x ^ 2)) (Set.Ici x) x

full_name: Real.hasDerivWithinAt_arcsin_Iic
code: theorem hasDerivWithinAt_arcsin_Iic {x : ℝ} (h : x ≠ 1) :
    HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Iic x) x 
goal: HasDerivWithinAt Real.arcsin (1 / √(1 - x ^ 2)) (Set.Iic x) x

full_name: Real.differentiableWithinAt_arcsin_Ici
code: theorem differentiableWithinAt_arcsin_Ici {x : ℝ} :
    DifferentiableWithinAt ℝ arcsin (Ici x) x ↔ x ≠ -1 
goal: DifferentiableWithinAt ℝ Real.arcsin (Set.Ici x) x ↔ x ≠ -1

full_name: Real.differentiableWithinAt_arcsin_Iic
code: theorem differentiableWithinAt_arcsin_Iic {x : ℝ} :
    DifferentiableWithinAt ℝ arcsin (Iic x) x ↔ x ≠ 1 
goal: DifferentiableWithinAt ℝ Real.arcsin (Set.Iic x) x ↔ x ≠ 1

full_name: Real.differentiableAt_arcsin
code: theorem differentiableAt_arcsin {x : ℝ} : DifferentiableAt ℝ arcsin x ↔ x ≠ -1 ∧ x ≠ 1 
goal: DifferentiableAt ℝ Real.arcsin x ↔ x ≠ -1 ∧ x ≠ 1

full_name: Real.deriv_arcsin
code: @[simp]
theorem deriv_arcsin : deriv arcsin = fun x => 1 / √(1 - x ^ 2) 
goal: deriv Real.arcsin = fun x => 1 / √(1 - x ^ 2)

full_name: Real.differentiableOn_arcsin
code: theorem differentiableOn_arcsin : DifferentiableOn ℝ arcsin {-1, 1}ᶜ 
goal: DifferentiableOn ℝ Real.arcsin {-1, 1}ᶜ

full_name: Real.contDiffOn_arcsin
code: theorem contDiffOn_arcsin {n : ℕ∞} : ContDiffOn ℝ n arcsin {-1, 1}ᶜ 
goal: ContDiffOn ℝ n Real.arcsin {-1, 1}ᶜ

full_name: Real.contDiffAt_arcsin_iff
code: theorem contDiffAt_arcsin_iff {x : ℝ} {n : ℕ∞} : ContDiffAt ℝ n arcsin x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1 
goal: ContDiffAt ℝ n Real.arcsin x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1

full_name: Real.hasStrictDerivAt_arccos
code: theorem hasStrictDerivAt_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasStrictDerivAt arccos (-(1 / √(1 - x ^ 2))) x 
goal: HasStrictDerivAt Real.arccos (-(1 / √(1 - x ^ 2))) x

full_name: Real.hasDerivAt_arccos
code: theorem hasDerivAt_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasDerivAt arccos (-(1 / √(1 - x ^ 2))) x 
goal: HasDerivAt Real.arccos (-(1 / √(1 - x ^ 2))) x

full_name: Real.contDiffAt_arccos
code: theorem contDiffAt_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞} : ContDiffAt ℝ n arccos x 
goal: ContDiffAt ℝ n Real.arccos x

full_name: Real.hasDerivWithinAt_arccos_Ici
code: theorem hasDerivWithinAt_arccos_Ici {x : ℝ} (h : x ≠ -1) :
    HasDerivWithinAt arccos (-(1 / √(1 - x ^ 2))) (Ici x) x 
goal: HasDerivWithinAt Real.arccos (-(1 / √(1 - x ^ 2))) (Set.Ici x) x

full_name: Real.hasDerivWithinAt_arccos_Iic
code: theorem hasDerivWithinAt_arccos_Iic {x : ℝ} (h : x ≠ 1) :
    HasDerivWithinAt arccos (-(1 / √(1 - x ^ 2))) (Iic x) x 
goal: HasDerivWithinAt Real.arccos (-(1 / √(1 - x ^ 2))) (Set.Iic x) x

full_name: Real.differentiableWithinAt_arccos_Ici
code: theorem differentiableWithinAt_arccos_Ici {x : ℝ} :
    DifferentiableWithinAt ℝ arccos (Ici x) x ↔ x ≠ -1 
goal: DifferentiableWithinAt ℝ Real.arccos (Set.Ici x) x ↔ x ≠ -1

full_name: Real.differentiableWithinAt_arccos_Iic
code: theorem differentiableWithinAt_arccos_Iic {x : ℝ} :
    DifferentiableWithinAt ℝ arccos (Iic x) x ↔ x ≠ 1 
goal: DifferentiableWithinAt ℝ Real.arccos (Set.Iic x) x ↔ x ≠ 1

full_name: Real.differentiableAt_arccos
code: theorem differentiableAt_arccos {x : ℝ} : DifferentiableAt ℝ arccos x ↔ x ≠ -1 ∧ x ≠ 1 
goal: DifferentiableAt ℝ Real.arccos x ↔ x ≠ -1 ∧ x ≠ 1

full_name: Real.deriv_arccos
code: @[simp]
theorem deriv_arccos : deriv arccos = fun x => -(1 / √(1 - x ^ 2)) 
goal: deriv Real.arccos = fun x => -(1 / √(1 - x ^ 2))

full_name: Real.differentiableOn_arccos
code: theorem differentiableOn_arccos : DifferentiableOn ℝ arccos {-1, 1}ᶜ 
goal: DifferentiableOn ℝ Real.arccos {-1, 1}ᶜ

full_name: Real.contDiffOn_arccos
code: theorem contDiffOn_arccos {n : ℕ∞} : ContDiffOn ℝ n arccos {-1, 1}ᶜ 
goal: ContDiffOn ℝ n Real.arccos {-1, 1}ᶜ

full_name: Real.contDiffAt_arccos_iff
code: theorem contDiffAt_arccos_iff {x : ℝ} {n : ℕ∞} :
    ContDiffAt ℝ n arccos x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1 
goal: ContDiffAt ℝ n Real.arccos x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1