ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasStrictFDerivAt.cpow
theorem HasStrictFDerivAt.cpow (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x)
    (h0 : f x ∈ slitPlane) : HasStrictFDerivAt (fun x => f x ^ g x)
      ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {f' g' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasStrictFDerivAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x
-- HasStrictFDerivAt.const_cpow
theorem HasStrictFDerivAt.const_cpow (hf : HasStrictFDerivAt f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    HasStrictFDerivAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {c : ℂ} (hf : HasStrictFDerivAt f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: HasStrictFDerivAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') x
-- HasFDerivAt.cpow
theorem HasFDerivAt.cpow (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x)
    (h0 : f x ∈ slitPlane) : HasFDerivAt (fun x => f x ^ g x)
      ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {f' g' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasFDerivAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x
-- HasFDerivAt.const_cpow
theorem HasFDerivAt.const_cpow (hf : HasFDerivAt f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    HasFDerivAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {c : ℂ} (hf : HasFDerivAt f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: HasFDerivAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') x
-- HasFDerivWithinAt.cpow
theorem HasFDerivWithinAt.cpow (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x)
    (h0 : f x ∈ slitPlane) : HasFDerivWithinAt (fun x => f x ^ g x)
      ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {f' g' : E →L[ℂ] ℂ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasFDerivWithinAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') s x
-- HasFDerivWithinAt.const_cpow
theorem HasFDerivWithinAt.const_cpow (hf : HasFDerivWithinAt f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    HasFDerivWithinAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {s : Set E} {c : ℂ} (hf : HasFDerivWithinAt f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: HasFDerivWithinAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') s x
-- DifferentiableAt.cpow
theorem DifferentiableAt.cpow (hf : DifferentiableAt ℂ f x) (hg : DifferentiableAt ℂ g x)
    (h0 : f x ∈ slitPlane) : DifferentiableAt ℂ (fun x => f x ^ g x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {x : E} (hf : DifferentiableAt ℂ f x) (hg : DifferentiableAt ℂ g x) (h0 : f x ∈ Complex.slitPlane)
Goal: DifferentiableAt ℂ (fun x => f x ^ g x) x
-- DifferentiableAt.const_cpow
theorem DifferentiableAt.const_cpow (hf : DifferentiableAt ℂ f x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    DifferentiableAt ℂ (fun x => c ^ f x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {c : ℂ} (hf : DifferentiableAt ℂ f x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: DifferentiableAt ℂ (fun x => c ^ f x) x
-- DifferentiableWithinAt.cpow
theorem DifferentiableWithinAt.cpow (hf : DifferentiableWithinAt ℂ f s x)
    (hg : DifferentiableWithinAt ℂ g s x) (h0 : f x ∈ slitPlane) :
    DifferentiableWithinAt ℂ (fun x => f x ^ g x) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) (hg : DifferentiableWithinAt ℂ g s x) (h0 : f x ∈ Complex.slitPlane)
Goal: DifferentiableWithinAt ℂ (fun x => f x ^ g x) s x
-- DifferentiableWithinAt.const_cpow
theorem DifferentiableWithinAt.const_cpow (hf : DifferentiableWithinAt ℂ f s x)
    (h0 : c ≠ 0 ∨ f x ≠ 0) : DifferentiableWithinAt ℂ (fun x => c ^ f x) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {c : ℂ} (hf : DifferentiableWithinAt ℂ f s x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: DifferentiableWithinAt ℂ (fun x => c ^ f x) s x
-- DifferentiableOn.cpow
theorem DifferentiableOn.cpow (hf : DifferentiableOn ℂ f s) (hg : DifferentiableOn ℂ g s)
    (h0 : Set.MapsTo f s slitPlane) : DifferentiableOn ℂ (fun x ↦ f x ^ g x) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {s : Set E} (hf : DifferentiableOn ℂ f s) (hg : DifferentiableOn ℂ g s) (h0 : Set.MapsTo f s Complex.slitPlane)
Goal: DifferentiableOn ℂ (fun x => f x ^ g x) s
-- DifferentiableOn.const_cpow
theorem DifferentiableOn.const_cpow (hf : DifferentiableOn ℂ f s)
    (h0 : c ≠ 0 ∨ ∀ x ∈ s, f x ≠ 0) : DifferentiableOn ℂ (fun x ↦ c ^ f x) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {c : ℂ} (hf : DifferentiableOn ℂ f s) (h0 : c ≠ 0 ∨ ∀ x ∈ s, f x ≠ 0)
Goal: DifferentiableOn ℂ (fun x => c ^ f x) s
-- Differentiable.cpow
theorem Differentiable.cpow (hf : Differentiable ℂ f) (hg : Differentiable ℂ g)
    (h0 : ∀ x, f x ∈ slitPlane) : Differentiable ℂ (fun x ↦ f x ^ g x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} (hf : Differentiable ℂ f) (hg : Differentiable ℂ g) (h0 : ∀ (x : E), f x ∈ Complex.slitPlane)
Goal: Differentiable ℂ fun x => f x ^ g x
-- Differentiable.const_cpow
theorem Differentiable.const_cpow (hf : Differentiable ℂ f)
    (h0 : c ≠ 0 ∨ ∀ x, f x ≠ 0) : Differentiable ℂ (fun x ↦ c ^ f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {c : ℂ} (hf : Differentiable ℂ f) (h0 : c ≠ 0 ∨ ∀ (x : E), f x ≠ 0)
Goal: Differentiable ℂ fun x => c ^ f x
-- _private.Mathlib.Analysis.SpecialFunctions.Pow.Deriv.0.aux
private theorem aux : ((g x * f x ^ (g x - 1)) • (1 : ℂ →L[ℂ] ℂ).smulRight f' +
    (f x ^ g x * log (f x)) • (1 : ℂ →L[ℂ] ℂ).smulRight g') 1 =
      g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g' 
Arguments: {f g : ℂ → ℂ} {f' g' x : ℂ}
Goal: ((g x * f x ^ (g x - 1)) • ContinuousLinearMap.smulRight 1 f' +
      (f x ^ g x * Complex.log (f x)) • ContinuousLinearMap.smulRight 1 g')
    1 =
  g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g'
-- HasStrictDerivAt.cpow
nonrec theorem HasStrictDerivAt.cpow (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x)
    (h0 : f x ∈ slitPlane) : HasStrictDerivAt (fun x => f x ^ g x)
      (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') x 
Arguments: {f g : ℂ → ℂ} {f' g' x : ℂ} (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasStrictDerivAt (fun x => f x ^ g x) (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') x
-- HasStrictDerivAt.const_cpow
theorem HasStrictDerivAt.const_cpow (hf : HasStrictDerivAt f f' x) (h : c ≠ 0 ∨ f x ≠ 0) :
    HasStrictDerivAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') x 
Arguments: {f : ℂ → ℂ} {f' x c : ℂ} (hf : HasStrictDerivAt f f' x) (h : c ≠ 0 ∨ f x ≠ 0)
Goal: HasStrictDerivAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') x
-- Complex.hasStrictDerivAt_cpow_const
theorem Complex.hasStrictDerivAt_cpow_const (h : x ∈ slitPlane) :
    HasStrictDerivAt (fun z : ℂ => z ^ c) (c * x ^ (c - 1)) x 
Arguments: {x c : ℂ} (h : x ∈ Complex.slitPlane)
Goal: HasStrictDerivAt (fun z => z ^ c) (c * x ^ (c - 1)) x
-- HasStrictDerivAt.cpow_const
theorem HasStrictDerivAt.cpow_const (hf : HasStrictDerivAt f f' x)
    (h0 : f x ∈ slitPlane) :
    HasStrictDerivAt (fun x => f x ^ c) (c * f x ^ (c - 1) * f') x 
Arguments: {f : ℂ → ℂ} {f' x c : ℂ} (hf : HasStrictDerivAt f f' x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasStrictDerivAt (fun x => f x ^ c) (c * f x ^ (c - 1) * f') x
-- HasDerivAt.cpow
theorem HasDerivAt.cpow (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x)
    (h0 : f x ∈ slitPlane) : HasDerivAt (fun x => f x ^ g x)
      (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') x 
Arguments: {f g : ℂ → ℂ} {f' g' x : ℂ} (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasDerivAt (fun x => f x ^ g x) (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') x
-- HasDerivAt.const_cpow
theorem HasDerivAt.const_cpow (hf : HasDerivAt f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    HasDerivAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') x 
Arguments: {f : ℂ → ℂ} {f' x c : ℂ} (hf : HasDerivAt f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: HasDerivAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') x
-- HasDerivAt.cpow_const
theorem HasDerivAt.cpow_const (hf : HasDerivAt f f' x) (h0 : f x ∈ slitPlane) :
    HasDerivAt (fun x => f x ^ c) (c * f x ^ (c - 1) * f') x 
Arguments: {f : ℂ → ℂ} {f' x c : ℂ} (hf : HasDerivAt f f' x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasDerivAt (fun x => f x ^ c) (c * f x ^ (c - 1) * f') x
-- HasDerivWithinAt.cpow
theorem HasDerivWithinAt.cpow (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
    (h0 : f x ∈ slitPlane) : HasDerivWithinAt (fun x => f x ^ g x)
      (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') s x 
Arguments: {f g : ℂ → ℂ} {s : Set ℂ} {f' g' x : ℂ} (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasDerivWithinAt (fun x => f x ^ g x) (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') s x
-- HasDerivWithinAt.const_cpow
theorem HasDerivWithinAt.const_cpow (hf : HasDerivWithinAt f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
    HasDerivWithinAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') s x 
Arguments: {f : ℂ → ℂ} {s : Set ℂ} {f' x c : ℂ} (hf : HasDerivWithinAt f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0)
Goal: HasDerivWithinAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') s x
-- HasDerivWithinAt.cpow_const
theorem HasDerivWithinAt.cpow_const (hf : HasDerivWithinAt f f' s x)
    (h0 : f x ∈ slitPlane) :
    HasDerivWithinAt (fun x => f x ^ c) (c * f x ^ (c - 1) * f') s x 
Arguments: {f : ℂ → ℂ} {s : Set ℂ} {f' x c : ℂ} (hf : HasDerivWithinAt f f' s x) (h0 : f x ∈ Complex.slitPlane)
Goal: HasDerivWithinAt (fun x => f x ^ c) (c * f x ^ (c - 1) * f') s x
-- hasDerivAt_ofReal_cpow
theorem hasDerivAt_ofReal_cpow {x : ℝ} (hx : x ≠ 0) {r : ℂ} (hr : r ≠ -1) :
    HasDerivAt (fun y : ℝ => (y : ℂ) ^ (r + 1) / (r + 1)) (x ^ r) x 
Arguments: {x : ℝ} (hx : x ≠ 0) {r : ℂ} (hr : r ≠ -1)
Goal: HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x
-- Real.hasStrictFDerivAt_rpow_of_pos
theorem hasStrictFDerivAt_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.1) :
    HasStrictFDerivAt (fun x : ℝ × ℝ => x.1 ^ x.2)
      ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
        (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ) p 
Arguments: (p : ℝ × ℝ) (hp : 0 < p.1)
Goal: HasStrictFDerivAt (fun x => x.1 ^ x.2)
  ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * Real.log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ)
  p
-- Real.hasStrictFDerivAt_rpow_of_neg
theorem hasStrictFDerivAt_rpow_of_neg (p : ℝ × ℝ) (hp : p.1 < 0) :
    HasStrictFDerivAt (fun x : ℝ × ℝ => x.1 ^ x.2)
      ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
        (p.1 ^ p.2 * log p.1 - exp (log p.1 * p.2) * sin (p.2 * π) * π) •
          ContinuousLinearMap.snd ℝ ℝ ℝ) p 
Arguments: (p : ℝ × ℝ) (hp : p.1 < 0)
Goal: HasStrictFDerivAt (fun x => x.1 ^ x.2)
  ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
    (p.1 ^ p.2 * Real.log p.1 - rexp (Real.log p.1 * p.2) * Real.sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ)
  p
-- Real.contDiffAt_rpow_of_ne
theorem contDiffAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) {n : ℕ∞} :
    ContDiffAt ℝ n (fun p : ℝ × ℝ => p.1 ^ p.2) p 
Arguments: (p : ℝ × ℝ) (hp : p.1 ≠ 0) {n : ℕ∞}
Goal: ContDiffAt ℝ n (fun p => p.1 ^ p.2) p
-- Real.differentiableAt_rpow_of_ne
theorem differentiableAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) :
    DifferentiableAt ℝ (fun p : ℝ × ℝ => p.1 ^ p.2) p 
Arguments: (p : ℝ × ℝ) (hp : p.1 ≠ 0)
Goal: DifferentiableAt ℝ (fun p => p.1 ^ p.2) p
-- HasStrictDerivAt.rpow
theorem _root_.HasStrictDerivAt.rpow {f g : ℝ → ℝ} {f' g' : ℝ} (hf : HasStrictDerivAt f f' x)
    (hg : HasStrictDerivAt g g' x) (h : 0 < f x) : HasStrictDerivAt (fun x => f x ^ g x)
      (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x 
Arguments: {x : ℝ} {f g : ℝ → ℝ} {f' g' : ℝ} (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x) (h : 0 < f x)
Goal: HasStrictDerivAt (fun x => f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x
-- Real.hasStrictDerivAt_rpow_const_of_ne
theorem hasStrictDerivAt_rpow_const_of_ne {x : ℝ} (hx : x ≠ 0) (p : ℝ) :
    HasStrictDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x 
Arguments: {x : ℝ} (hx : x ≠ 0) (p : ℝ)
Goal: HasStrictDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x
-- Real.hasStrictDerivAt_const_rpow
theorem hasStrictDerivAt_const_rpow {a : ℝ} (ha : 0 < a) (x : ℝ) :
    HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a) x 
Arguments: {a : ℝ} (ha : 0 < a) (x : ℝ)
Goal: HasStrictDerivAt (fun x => a ^ x) (a ^ x * Real.log a) x
-- Real.differentiableAt_rpow_const_of_ne
lemma differentiableAt_rpow_const_of_ne (p : ℝ) {x : ℝ} (hx : x ≠ 0) :
    DifferentiableAt ℝ (fun x => x ^ p) x 
Arguments: (p : ℝ) {x : ℝ} (hx : x ≠ 0)
Goal: DifferentiableAt ℝ (fun x => x ^ p) x
-- Real.differentiableOn_rpow_const
lemma differentiableOn_rpow_const (p : ℝ) :
    DifferentiableOn ℝ (fun x => (x : ℝ) ^ p) {0}ᶜ 
Arguments: (p : ℝ)
Goal: DifferentiableOn ℝ (fun x => x ^ p) {0}ᶜ
-- Real.hasStrictDerivAt_const_rpow_of_neg
theorem hasStrictDerivAt_const_rpow_of_neg {a x : ℝ} (ha : a < 0) :
    HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a - exp (log a * x) * sin (x * π) * π) x 
Arguments: {a x : ℝ} (ha : a < 0)
Goal: HasStrictDerivAt (fun x => a ^ x) (a ^ x * Real.log a - rexp (Real.log a * x) * Real.sin (x * π) * π) x
-- Real.hasDerivAt_rpow_const
theorem hasDerivAt_rpow_const {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p) :
    HasDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x 
Arguments: {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p)
Goal: HasDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x
-- Real.differentiable_rpow_const
theorem differentiable_rpow_const {p : ℝ} (hp : 1 ≤ p) : Differentiable ℝ fun x : ℝ => x ^ p 
Arguments: {p : ℝ} (hp : 1 ≤ p)
Goal: Differentiable ℝ fun x => x ^ p
-- Real.deriv_rpow_const
theorem deriv_rpow_const {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p) :
    deriv (fun x : ℝ => x ^ p) x = p * x ^ (p - 1) 
Arguments: {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p)
Goal: deriv (fun x => x ^ p) x = p * x ^ (p - 1)
-- Real.deriv_rpow_const'
theorem deriv_rpow_const' {p : ℝ} (h : 1 ≤ p) :
    (deriv fun x : ℝ => x ^ p) = fun x => p * x ^ (p - 1) 
Arguments: {p : ℝ} (h : 1 ≤ p)
Goal: (deriv fun x => x ^ p) = fun x => p * x ^ (p - 1)
-- Real.contDiffAt_rpow_const_of_ne
theorem contDiffAt_rpow_const_of_ne {x p : ℝ} {n : ℕ∞} (h : x ≠ 0) :
    ContDiffAt ℝ n (fun x => x ^ p) x 
Arguments: {x p : ℝ} {n : ℕ∞} (h : x ≠ 0)
Goal: ContDiffAt ℝ n (fun x => x ^ p) x
-- Real.contDiff_rpow_const_of_le
theorem contDiff_rpow_const_of_le {p : ℝ} {n : ℕ} (h : ↑n ≤ p) :
    ContDiff ℝ n fun x : ℝ => x ^ p 
Arguments: {p : ℝ} {n : ℕ} (h : ↑n ≤ p)
Goal: ContDiff ℝ ↑n fun x => x ^ p
-- Real.contDiffAt_rpow_const_of_le
theorem contDiffAt_rpow_const_of_le {x p : ℝ} {n : ℕ} (h : ↑n ≤ p) :
    ContDiffAt ℝ n (fun x : ℝ => x ^ p) x 
Arguments: {x p : ℝ} {n : ℕ} (h : ↑n ≤ p)
Goal: ContDiffAt ℝ (↑n) (fun x => x ^ p) x
-- Real.contDiffAt_rpow_const
theorem contDiffAt_rpow_const {x p : ℝ} {n : ℕ} (h : x ≠ 0 ∨ ↑n ≤ p) :
    ContDiffAt ℝ n (fun x : ℝ => x ^ p) x 
Arguments: {x p : ℝ} {n : ℕ} (h : x ≠ 0 ∨ ↑n ≤ p)
Goal: ContDiffAt ℝ (↑n) (fun x => x ^ p) x
-- Real.hasStrictDerivAt_rpow_const
theorem hasStrictDerivAt_rpow_const {x p : ℝ} (hx : x ≠ 0 ∨ 1 ≤ p) :
    HasStrictDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x 
Arguments: {x p : ℝ} (hx : x ≠ 0 ∨ 1 ≤ p)
Goal: HasStrictDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x
-- HasFDerivWithinAt.rpow
theorem HasFDerivWithinAt.rpow (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x)
    (h : 0 < f x) : HasFDerivWithinAt (fun x => f x ^ g x)
      ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {f' g' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x) (h : 0 < f x)
Goal: HasFDerivWithinAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') s x
-- HasFDerivAt.rpow
theorem HasFDerivAt.rpow (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) (h : 0 < f x) :
    HasFDerivAt (fun x => f x ^ g x)
      ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {f' g' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) (h : 0 < f x)
Goal: HasFDerivAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') x
-- HasStrictFDerivAt.rpow
theorem HasStrictFDerivAt.rpow (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x)
    (h : 0 < f x) : HasStrictFDerivAt (fun x => f x ^ g x)
      ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {f' g' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) (h : 0 < f x)
Goal: HasStrictFDerivAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') x
-- DifferentiableWithinAt.rpow
theorem DifferentiableWithinAt.rpow (hf : DifferentiableWithinAt ℝ f s x)
    (hg : DifferentiableWithinAt ℝ g s x) (h : f x ≠ 0) :
    DifferentiableWithinAt ℝ (fun x => f x ^ g x) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hg : DifferentiableWithinAt ℝ g s x) (h : f x ≠ 0)
Goal: DifferentiableWithinAt ℝ (fun x => f x ^ g x) s x
-- DifferentiableAt.rpow
theorem DifferentiableAt.rpow (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
    (h : f x ≠ 0) : DifferentiableAt ℝ (fun x => f x ^ g x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (h : f x ≠ 0)
Goal: DifferentiableAt ℝ (fun x => f x ^ g x) x
-- DifferentiableOn.rpow
theorem DifferentiableOn.rpow (hf : DifferentiableOn ℝ f s) (hg : DifferentiableOn ℝ g s)
    (h : ∀ x ∈ s, f x ≠ 0) : DifferentiableOn ℝ (fun x => f x ^ g x) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {s : Set E} (hf : DifferentiableOn ℝ f s) (hg : DifferentiableOn ℝ g s) (h : ∀ x ∈ s, f x ≠ 0)
Goal: DifferentiableOn ℝ (fun x => f x ^ g x) s
-- Differentiable.rpow
theorem Differentiable.rpow (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (h : ∀ x, f x ≠ 0) :
    Differentiable ℝ fun x => f x ^ g x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (h : ∀ (x : E), f x ≠ 0)
Goal: Differentiable ℝ fun x => f x ^ g x
-- HasFDerivWithinAt.rpow_const
theorem HasFDerivWithinAt.rpow_const (hf : HasFDerivWithinAt f f' s x) (h : f x ≠ 0 ∨ 1 ≤ p) :
    HasFDerivWithinAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} {p : ℝ} (hf : HasFDerivWithinAt f f' s x) (h : f x ≠ 0 ∨ 1 ≤ p)
Goal: HasFDerivWithinAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') s x
-- HasFDerivAt.rpow_const
theorem HasFDerivAt.rpow_const (hf : HasFDerivAt f f' x) (h : f x ≠ 0 ∨ 1 ≤ p) :
    HasFDerivAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {p : ℝ} (hf : HasFDerivAt f f' x) (h : f x ≠ 0 ∨ 1 ≤ p)
Goal: HasFDerivAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') x
-- HasStrictFDerivAt.rpow_const
theorem HasStrictFDerivAt.rpow_const (hf : HasStrictFDerivAt f f' x) (h : f x ≠ 0 ∨ 1 ≤ p) :
    HasStrictFDerivAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {p : ℝ} (hf : HasStrictFDerivAt f f' x) (h : f x ≠ 0 ∨ 1 ≤ p)
Goal: HasStrictFDerivAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') x
-- DifferentiableWithinAt.rpow_const
theorem DifferentiableWithinAt.rpow_const (hf : DifferentiableWithinAt ℝ f s x)
    (h : f x ≠ 0 ∨ 1 ≤ p) : DifferentiableWithinAt ℝ (fun x => f x ^ p) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {p : ℝ} (hf : DifferentiableWithinAt ℝ f s x) (h : f x ≠ 0 ∨ 1 ≤ p)
Goal: DifferentiableWithinAt ℝ (fun x => f x ^ p) s x
-- DifferentiableAt.rpow_const
@[simp]
theorem DifferentiableAt.rpow_const (hf : DifferentiableAt ℝ f x) (h : f x ≠ 0 ∨ 1 ≤ p) :
    DifferentiableAt ℝ (fun x => f x ^ p) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {p : ℝ} (hf : DifferentiableAt ℝ f x) (h : f x ≠ 0 ∨ 1 ≤ p)
Goal: DifferentiableAt ℝ (fun x => f x ^ p) x
-- DifferentiableOn.rpow_const
theorem DifferentiableOn.rpow_const (hf : DifferentiableOn ℝ f s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 1 ≤ p) :
    DifferentiableOn ℝ (fun x => f x ^ p) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {p : ℝ} (hf : DifferentiableOn ℝ f s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 1 ≤ p)
Goal: DifferentiableOn ℝ (fun x => f x ^ p) s
-- Differentiable.rpow_const
theorem Differentiable.rpow_const (hf : Differentiable ℝ f) (h : ∀ x, f x ≠ 0 ∨ 1 ≤ p) :
    Differentiable ℝ fun x => f x ^ p 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {p : ℝ} (hf : Differentiable ℝ f) (h : ∀ (x : E), f x ≠ 0 ∨ 1 ≤ p)
Goal: Differentiable ℝ fun x => f x ^ p
-- HasFDerivWithinAt.const_rpow
theorem HasFDerivWithinAt.const_rpow (hf : HasFDerivWithinAt f f' s x) (hc : 0 < c) :
    HasFDerivWithinAt (fun x => c ^ f x) ((c ^ f x * Real.log c) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} {c : ℝ} (hf : HasFDerivWithinAt f f' s x) (hc : 0 < c)
Goal: HasFDerivWithinAt (fun x => c ^ f x) ((c ^ f x * Real.log c) • f') s x
-- HasFDerivAt.const_rpow
theorem HasFDerivAt.const_rpow (hf : HasFDerivAt f f' x) (hc : 0 < c) :
    HasFDerivAt (fun x => c ^ f x) ((c ^ f x * Real.log c) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {c : ℝ} (hf : HasFDerivAt f f' x) (hc : 0 < c)
Goal: HasFDerivAt (fun x => c ^ f x) ((c ^ f x * Real.log c) • f') x
-- HasStrictFDerivAt.const_rpow
theorem HasStrictFDerivAt.const_rpow (hf : HasStrictFDerivAt f f' x) (hc : 0 < c) :
    HasStrictFDerivAt (fun x => c ^ f x) ((c ^ f x * Real.log c) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {c : ℝ} (hf : HasStrictFDerivAt f f' x) (hc : 0 < c)
Goal: HasStrictFDerivAt (fun x => c ^ f x) ((c ^ f x * Real.log c) • f') x
-- ContDiffWithinAt.rpow
theorem ContDiffWithinAt.rpow (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
    (h : f x ≠ 0) : ContDiffWithinAt ℝ n (fun x => f x ^ g x) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x) (h : f x ≠ 0)
Goal: ContDiffWithinAt ℝ n (fun x => f x ^ g x) s x