File: Mathlib/Analysis/Calculus/Deriv/ZPow.lean

full_name: hasStrictDerivAt_zpow
code: theorem hasStrictDerivAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
    HasStrictDerivAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) x 
goal: HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x

full_name: hasDerivAt_zpow
code: theorem hasDerivAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
    HasDerivAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) x 
goal: HasDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x

full_name: hasDerivWithinAt_zpow
code: theorem hasDerivWithinAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) s x 
goal: HasDerivWithinAt (fun x => x ^ m) (↑m * x ^ (m - 1)) s x

full_name: differentiableAt_zpow
code: theorem differentiableAt_zpow : DifferentiableAt 𝕜 (fun x => x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m 
goal: DifferentiableAt 𝕜 (fun x => x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m

full_name: differentiableWithinAt_zpow
code: theorem differentiableWithinAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
    DifferentiableWithinAt 𝕜 (fun x => x ^ m) s x 
goal: DifferentiableWithinAt 𝕜 (fun x => x ^ m) s x

full_name: differentiableOn_zpow
code: theorem differentiableOn_zpow (m : ℤ) (s : Set 𝕜) (h : (0 : 𝕜) ∉ s ∨ 0 ≤ m) :
    DifferentiableOn 𝕜 (fun x => x ^ m) s 
goal: DifferentiableOn 𝕜 (fun x => x ^ m) s

full_name: deriv_zpow
code: theorem deriv_zpow (m : ℤ) (x : 𝕜) : deriv (fun x => x ^ m) x = m * x ^ (m - 1) 
goal: deriv (fun x => x ^ m) x = ↑m * x ^ (m - 1)

full_name: deriv_zpow'
code: @[simp]
theorem deriv_zpow' (m : ℤ) : (deriv fun x : 𝕜 => x ^ m) = fun x => (m : 𝕜) * x ^ (m - 1) 
goal: (deriv fun x => x ^ m) = fun x => ↑m * x ^ (m - 1)

full_name: derivWithin_zpow
code: theorem derivWithin_zpow (hxs : UniqueDiffWithinAt 𝕜 s x) (h : x ≠ 0 ∨ 0 ≤ m) :
    derivWithin (fun x => x ^ m) s x = (m : 𝕜) * x ^ (m - 1) 
goal: derivWithin (fun x => x ^ m) s x = ↑m * x ^ (m - 1)

full_name: iter_deriv_zpow'
code: @[simp]
theorem iter_deriv_zpow' (m : ℤ) (k : ℕ) :
    (deriv^[k] fun x : 𝕜 => x ^ m) =
      fun x => (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k) 
goal: (deriv^[k] fun x => x ^ m) = fun x => (∏ i ∈ Finset.range k, (↑m - ↑i)) * x ^ (m - ↑k)

full_name: iter_deriv_zpow
code: theorem iter_deriv_zpow (m : ℤ) (x : 𝕜) (k : ℕ) :
    deriv^[k] (fun y => y ^ m) x = (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k) 
goal: deriv^[k] (fun y => y ^ m) x = (∏ i ∈ Finset.range k, (↑m - ↑i)) * x ^ (m - ↑k)

full_name: iter_deriv_pow
code: theorem iter_deriv_pow (n : ℕ) (x : 𝕜) (k : ℕ) :
    deriv^[k] (fun x : 𝕜 => x ^ n) x = (∏ i ∈ Finset.range k, ((n : 𝕜) - i)) * x ^ (n - k) 
goal: deriv^[k] (fun x => x ^ n) x = (∏ i ∈ Finset.range k, (↑n - ↑i)) * x ^ (n - k)

full_name: iter_deriv_pow'
code: @[simp]
theorem iter_deriv_pow' (n k : ℕ) :
    (deriv^[k] fun x : 𝕜 => x ^ n) =
      fun x => (∏ i ∈ Finset.range k, ((n : 𝕜) - i)) * x ^ (n - k) 
goal: (deriv^[k] fun x => x ^ n) = fun x => (∏ i ∈ Finset.range k, (↑n - ↑i)) * x ^ (n - k)

full_name: iter_deriv_inv
code: theorem iter_deriv_inv (k : ℕ) (x : 𝕜) :
    deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - i : 𝕜)) * x ^ (-1 - k : ℤ) 
goal: deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - ↑i)) * x ^ (-1 - ↑k)

full_name: iter_deriv_inv'
code: @[simp]
theorem iter_deriv_inv' (k : ℕ) :
    deriv^[k] Inv.inv = fun x : 𝕜 => (∏ i ∈ Finset.range k, (-1 - i : 𝕜)) * x ^ (-1 - k : ℤ) 
goal: deriv^[k] Inv.inv = fun x => (∏ i ∈ Finset.range k, (-1 - ↑i)) * x ^ (-1 - ↑k)

full_name: DifferentiableWithinAt.zpow
code: theorem DifferentiableWithinAt.zpow (hf : DifferentiableWithinAt 𝕜 f t a) (h : f a ≠ 0 ∨ 0 ≤ m) :
    DifferentiableWithinAt 𝕜 (fun x => f x ^ m) t a 
goal: DifferentiableWithinAt 𝕜 (fun x => f x ^ m) t a

full_name: DifferentiableAt.zpow
code: theorem DifferentiableAt.zpow (hf : DifferentiableAt 𝕜 f a) (h : f a ≠ 0 ∨ 0 ≤ m) :
    DifferentiableAt 𝕜 (fun x => f x ^ m) a 
goal: DifferentiableAt 𝕜 (fun x => f x ^ m) a

full_name: DifferentiableOn.zpow
code: theorem DifferentiableOn.zpow (hf : DifferentiableOn 𝕜 f t) (h : (∀ x ∈ t, f x ≠ 0) ∨ 0 ≤ m) :
    DifferentiableOn 𝕜 (fun x => f x ^ m) t 
goal: DifferentiableOn 𝕜 (fun x => f x ^ m) t

full_name: Differentiable.zpow
code: theorem Differentiable.zpow (hf : Differentiable 𝕜 f) (h : (∀ x, f x ≠ 0) ∨ 0 ≤ m) :
    Differentiable 𝕜 fun x => f x ^ m 
goal: Differentiable 𝕜 fun x => f x ^ m