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

full_name: Real.hasStrictDerivAt_tan
code: theorem hasStrictDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x 
goal: HasStrictDerivAt Real.tan (1 / Real.cos x ^ 2) x

full_name: Real.hasDerivAt_tan
code: theorem hasDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasDerivAt tan (1 / cos x ^ 2) x 
goal: HasDerivAt Real.tan (1 / Real.cos x ^ 2) x

full_name: Real.tendsto_abs_tan_of_cos_eq_zero
code: theorem tendsto_abs_tan_of_cos_eq_zero {x : ℝ} (hx : cos x = 0) :
    Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop 
goal: Filter.Tendsto (fun x => |Real.tan x|) (nhdsWithin x {x}ᶜ) Filter.atTop

full_name: Real.tendsto_abs_tan_atTop
code: theorem tendsto_abs_tan_atTop (k : ℤ) :
    Tendsto (fun x => abs (tan x)) (𝓝[≠] ((2 * k + 1) * π / 2)) atTop 
goal: Filter.Tendsto (fun x => |Real.tan x|) (nhdsWithin ((2 * ↑k + 1) * Real.pi / 2) {(2 * ↑k + 1) * Real.pi / 2}ᶜ)
  Filter.atTop

full_name: Real.continuousAt_tan
code: theorem continuousAt_tan {x : ℝ} : ContinuousAt tan x ↔ cos x ≠ 0 
goal: ContinuousAt Real.tan x ↔ Real.cos x ≠ 0

full_name: Real.differentiableAt_tan
code: theorem differentiableAt_tan {x : ℝ} : DifferentiableAt ℝ tan x ↔ cos x ≠ 0 
goal: DifferentiableAt ℝ Real.tan x ↔ Real.cos x ≠ 0

full_name: Real.deriv_tan
code: @[simp]
theorem deriv_tan (x : ℝ) : deriv tan x = 1 / cos x ^ 2 
goal: deriv Real.tan x = 1 / Real.cos x ^ 2

full_name: Real.contDiffAt_tan
code: @[simp]
theorem contDiffAt_tan {n x} : ContDiffAt ℝ n tan x ↔ cos x ≠ 0 
goal: ContDiffAt ℝ n Real.tan x ↔ Real.cos x ≠ 0

full_name: Real.hasDerivAt_tan_of_mem_Ioo
code: theorem hasDerivAt_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π / 2) : ℝ) (π / 2)) :
    HasDerivAt tan (1 / cos x ^ 2) x 
goal: HasDerivAt Real.tan (1 / Real.cos x ^ 2) x

full_name: Real.differentiableAt_tan_of_mem_Ioo
code: theorem differentiableAt_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π / 2) : ℝ) (π / 2)) :
    DifferentiableAt ℝ tan x 
goal: DifferentiableAt ℝ Real.tan x

full_name: Real.hasStrictDerivAt_arctan
code: theorem hasStrictDerivAt_arctan (x : ℝ) : HasStrictDerivAt arctan (1 / (1 + x ^ 2)) x 
goal: HasStrictDerivAt Real.arctan (1 / (1 + x ^ 2)) x

full_name: Real.hasDerivAt_arctan
code: theorem hasDerivAt_arctan (x : ℝ) : HasDerivAt arctan (1 / (1 + x ^ 2)) x 
goal: HasDerivAt Real.arctan (1 / (1 + x ^ 2)) x

full_name: Real.hasDerivAt_arctan'
code: theorem hasDerivAt_arctan' (x : ℝ) : HasDerivAt arctan (1 + x ^ 2)⁻¹ x 
goal: HasDerivAt Real.arctan (1 + x ^ 2)⁻¹ x

full_name: Real.differentiableAt_arctan
code: theorem differentiableAt_arctan (x : ℝ) : DifferentiableAt ℝ arctan x 
goal: DifferentiableAt ℝ Real.arctan x

full_name: Real.differentiable_arctan
code: theorem differentiable_arctan : Differentiable ℝ arctan 
goal: Differentiable ℝ Real.arctan

full_name: Real.deriv_arctan
code: @[simp]
theorem deriv_arctan : deriv arctan = fun (x : ℝ) => 1 / (1 + x ^ 2) 
goal: deriv Real.arctan = fun x => 1 / (1 + x ^ 2)

full_name: Real.contDiff_arctan
code: theorem contDiff_arctan {n : ℕ∞} : ContDiff ℝ n arctan 
goal: ContDiff ℝ n Real.arctan

full_name: HasStrictDerivAt.arctan
code: theorem HasStrictDerivAt.arctan (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') x 
goal: HasStrictDerivAt (fun x => Real.arctan (f x)) (1 / (1 + f x ^ 2) * f') x

full_name: HasDerivAt.arctan
code: theorem HasDerivAt.arctan (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') x 
goal: HasDerivAt (fun x => Real.arctan (f x)) (1 / (1 + f x ^ 2) * f') x

full_name: HasDerivWithinAt.arctan
code: theorem HasDerivWithinAt.arctan (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') s x 
goal: HasDerivWithinAt (fun x => Real.arctan (f x)) (1 / (1 + f x ^ 2) * f') s x

full_name: derivWithin_arctan
code: theorem derivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => arctan (f x)) s x = 1 / (1 + f x ^ 2) * derivWithin f s x 
goal: derivWithin (fun x => Real.arctan (f x)) s x = 1 / (1 + f x ^ 2) * derivWithin f s x

full_name: deriv_arctan
code: @[simp]
theorem deriv_arctan (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => arctan (f x)) x = 1 / (1 + f x ^ 2) * deriv f x 
goal: deriv (fun x => Real.arctan (f x)) x = 1 / (1 + f x ^ 2) * deriv f x

full_name: HasStrictFDerivAt.arctan
code: theorem HasStrictFDerivAt.arctan (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x 
goal: HasStrictFDerivAt (fun x => Real.arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x

full_name: HasFDerivAt.arctan
code: theorem HasFDerivAt.arctan (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x 
goal: HasFDerivAt (fun x => Real.arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x

full_name: HasFDerivWithinAt.arctan
code: theorem HasFDerivWithinAt.arctan (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • f') s x 
goal: HasFDerivWithinAt (fun x => Real.arctan (f x)) ((1 / (1 + f x ^ 2)) • f') s x

full_name: fderivWithin_arctan
code: theorem fderivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => arctan (f x)) s x = (1 / (1 + f x ^ 2)) • fderivWithin ℝ f s x 
goal: fderivWithin ℝ (fun x => Real.arctan (f x)) s x = (1 / (1 + f x ^ 2)) • fderivWithin ℝ f s x

full_name: fderiv_arctan
code: @[simp]
theorem fderiv_arctan (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => arctan (f x)) x = (1 / (1 + f x ^ 2)) • fderiv ℝ f x 
goal: fderiv ℝ (fun x => Real.arctan (f x)) x = (1 / (1 + f x ^ 2)) • fderiv ℝ f x

full_name: DifferentiableWithinAt.arctan
code: theorem DifferentiableWithinAt.arctan (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.arctan (f x)) s x 
goal: DifferentiableWithinAt ℝ (fun x => Real.arctan (f x)) s x

full_name: DifferentiableAt.arctan
code: @[simp]
theorem DifferentiableAt.arctan (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => arctan (f x)) x 
goal: DifferentiableAt ℝ (fun x => Real.arctan (f x)) x

full_name: DifferentiableOn.arctan
code: theorem DifferentiableOn.arctan (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => arctan (f x)) s 
goal: DifferentiableOn ℝ (fun x => Real.arctan (f x)) s

full_name: Differentiable.arctan
code: @[simp]
theorem Differentiable.arctan (hc : Differentiable ℝ f) : Differentiable ℝ fun x => arctan (f x) 
goal: Differentiable ℝ fun x => Real.arctan (f x)

full_name: ContDiffAt.arctan
code: theorem ContDiffAt.arctan (h : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => arctan (f x)) x 
goal: ContDiffAt ℝ n (fun x => Real.arctan (f x)) x

full_name: ContDiff.arctan
code: theorem ContDiff.arctan (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => arctan (f x) 
goal: ContDiff ℝ n fun x => Real.arctan (f x)

full_name: ContDiffWithinAt.arctan
code: theorem ContDiffWithinAt.arctan (h : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => arctan (f x)) s x 
goal: ContDiffWithinAt ℝ n (fun x => Real.arctan (f x)) s x

full_name: ContDiffOn.arctan
code: theorem ContDiffOn.arctan (h : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => arctan (f x)) s 
goal: ContDiffOn ℝ n (fun x => Real.arctan (f x)) s