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

full_name: hasStrictDerivAt_inv
code: theorem hasStrictDerivAt_inv (hx : x ≠ 0) : HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x 
goal: HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x

full_name: hasDerivAt_inv
code: theorem hasDerivAt_inv (x_ne_zero : x ≠ 0) : HasDerivAt (fun y => y⁻¹) (-(x ^ 2)⁻¹) x 
goal: HasDerivAt (fun y => y⁻¹) (-(x ^ 2)⁻¹) x

full_name: hasDerivWithinAt_inv
code: theorem hasDerivWithinAt_inv (x_ne_zero : x ≠ 0) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => x⁻¹) (-(x ^ 2)⁻¹) s x 
goal: HasDerivWithinAt (fun x => x⁻¹) (-(x ^ 2)⁻¹) s x

full_name: differentiableAt_inv
code: theorem differentiableAt_inv : DifferentiableAt 𝕜 (fun x => x⁻¹) x ↔ x ≠ 0 
goal: DifferentiableAt 𝕜 (fun x => x⁻¹) x ↔ x ≠ 0

full_name: differentiableWithinAt_inv
code: theorem differentiableWithinAt_inv (x_ne_zero : x ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x 
goal: DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x

full_name: differentiableOn_inv
code: theorem differentiableOn_inv : DifferentiableOn 𝕜 (fun x : 𝕜 => x⁻¹) { x | x ≠ 0 } 
goal: DifferentiableOn 𝕜 (fun x => x⁻¹) {x | x ≠ 0}

full_name: deriv_inv
code: theorem deriv_inv : deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹ 
goal: deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹

full_name: deriv_inv'
code: @[simp]
theorem deriv_inv' : (deriv fun x : 𝕜 => x⁻¹) = fun x => -(x ^ 2)⁻¹ 
goal: (deriv fun x => x⁻¹) = fun x => -(x ^ 2)⁻¹

full_name: derivWithin_inv
code: theorem derivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => x⁻¹) s x = -(x ^ 2)⁻¹ 
goal: derivWithin (fun x => x⁻¹) s x = -(x ^ 2)⁻¹

full_name: hasFDerivAt_inv
code: theorem hasFDerivAt_inv (x_ne_zero : x ≠ 0) :
    HasFDerivAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) x 
goal: HasFDerivAt (fun x => x⁻¹) (ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)) x

full_name: hasFDerivWithinAt_inv
code: theorem hasFDerivWithinAt_inv (x_ne_zero : x ≠ 0) :
    HasFDerivWithinAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) s x 
goal: HasFDerivWithinAt (fun x => x⁻¹) (ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)) s x

full_name: fderiv_inv
code: theorem fderiv_inv : fderiv 𝕜 (fun x => x⁻¹) x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) 
goal: fderiv 𝕜 (fun x => x⁻¹) x = ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)

full_name: fderivWithin_inv
code: theorem fderivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => x⁻¹) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) 
goal: fderivWithin 𝕜 (fun x => x⁻¹) s x = ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)

full_name: HasDerivWithinAt.inv
code: theorem HasDerivWithinAt.inv (hc : HasDerivWithinAt c c' s x) (hx : c x ≠ 0) :
    HasDerivWithinAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) s x 
goal: HasDerivWithinAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) s x

full_name: HasDerivAt.inv
code: theorem HasDerivAt.inv (hc : HasDerivAt c c' x) (hx : c x ≠ 0) :
    HasDerivAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) x 
goal: HasDerivAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) x

full_name: DifferentiableWithinAt.inv
code: theorem DifferentiableWithinAt.inv (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z 
goal: DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z

full_name: DifferentiableAt.inv
code: @[simp]
theorem DifferentiableAt.inv (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0) :
    DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z 
goal: DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z

full_name: DifferentiableOn.inv
code: theorem DifferentiableOn.inv (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
    DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S 
goal: DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S

full_name: Differentiable.inv
code: @[simp]
theorem Differentiable.inv (hf : Differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
    Differentiable 𝕜 fun x => (h x)⁻¹ 
goal: Differentiable 𝕜 fun x => (h x)⁻¹

full_name: derivWithin_inv'
code: theorem derivWithin_inv' (hc : DifferentiableWithinAt 𝕜 c s x) (hx : c x ≠ 0)
    (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2 
goal: derivWithin (fun x => (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2

full_name: deriv_inv''
code: @[simp]
theorem deriv_inv'' (hc : DifferentiableAt 𝕜 c x) (hx : c x ≠ 0) :
    deriv (fun x => (c x)⁻¹) x = -deriv c x / c x ^ 2 
goal: deriv (fun x => (c x)⁻¹) x = -deriv c x / c x ^ 2

full_name: HasDerivWithinAt.div
code: theorem HasDerivWithinAt.div (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x)
    (hx : d x ≠ 0) :
    HasDerivWithinAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) s x 
goal: HasDerivWithinAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) s x

full_name: HasStrictDerivAt.div
code: theorem HasStrictDerivAt.div (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x)
    (hx : d x ≠ 0) : HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x 
goal: HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x

full_name: HasDerivAt.div
code: theorem HasDerivAt.div (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) (hx : d x ≠ 0) :
    HasDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x 
goal: HasDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x

full_name: DifferentiableWithinAt.div
code: theorem DifferentiableWithinAt.div (hc : DifferentiableWithinAt 𝕜 c s x)
    (hd : DifferentiableWithinAt 𝕜 d s x) (hx : d x ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => c x / d x) s x 
goal: DifferentiableWithinAt 𝕜 (fun x => c x / d x) s x

full_name: DifferentiableAt.div
code: @[simp]
theorem DifferentiableAt.div (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x)
    (hx : d x ≠ 0) : DifferentiableAt 𝕜 (fun x => c x / d x) x 
goal: DifferentiableAt 𝕜 (fun x => c x / d x) x

full_name: DifferentiableOn.div
code: theorem DifferentiableOn.div (hc : DifferentiableOn 𝕜 c s) (hd : DifferentiableOn 𝕜 d s)
    (hx : ∀ x ∈ s, d x ≠ 0) : DifferentiableOn 𝕜 (fun x => c x / d x) s 
goal: DifferentiableOn 𝕜 (fun x => c x / d x) s

full_name: Differentiable.div
code: @[simp]
theorem Differentiable.div (hc : Differentiable 𝕜 c) (hd : Differentiable 𝕜 d) (hx : ∀ x, d x ≠ 0) :
    Differentiable 𝕜 fun x => c x / d x 
goal: Differentiable 𝕜 fun x => c x / d x

full_name: derivWithin_div
code: theorem derivWithin_div (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x)
    (hx : d x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => c x / d x) s x =
      (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2 
goal: derivWithin (fun x => c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2

full_name: deriv_div
code: @[simp]
theorem deriv_div (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) (hx : d x ≠ 0) :
    deriv (fun x => c x / d x) x = (deriv c x * d x - c x * deriv d x) / d x ^ 2 
goal: deriv (fun x => c x / d x) x = (deriv c x * d x - c x * deriv d x) / d x ^ 2