File: Mathlib/Analysis/Calculus/Dslope.lean

full_name: dslope
code: /-- `dslope f a b` is defined as `slope f a b = (b - a)⁻¹ • (f b - f a)` for `a ≠ b` and
`deriv f a` for `a = b`. -/
noncomputable def dslope (f : 𝕜 → E) (a : 𝕜) : 𝕜 → E 
goal: 𝕜 → E

full_name: dslope_same
code: @[simp]
theorem dslope_same (f : 𝕜 → E) (a : 𝕜) : dslope f a a = deriv f a 
goal: dslope f a a = deriv f a

full_name: dslope_of_ne
code: theorem dslope_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope f a b = slope f a b 
goal: dslope f a b = slope f a b

full_name: ContinuousLinearMap.dslope_comp
code: theorem ContinuousLinearMap.dslope_comp {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
    (f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a) :
    dslope (f ∘ g) a b = f (dslope g a b) 
goal: dslope (⇑f ∘ g) a b = f (dslope g a b)

full_name: eqOn_dslope_slope
code: theorem eqOn_dslope_slope (f : 𝕜 → E) (a : 𝕜) : EqOn (dslope f a) (slope f a) {a}ᶜ 
goal: Set.EqOn (dslope f a) (slope f a) {a}ᶜ

full_name: dslope_eventuallyEq_slope_of_ne
code: theorem dslope_eventuallyEq_slope_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope f a =ᶠ[𝓝 b] slope f a 
goal: dslope f a =ᶠ[𝓝 b] slope f a

full_name: dslope_eventuallyEq_slope_punctured_nhds
code: theorem dslope_eventuallyEq_slope_punctured_nhds (f : 𝕜 → E) : dslope f a =ᶠ[𝓝[≠] a] slope f a 
goal: dslope f a =ᶠ[𝓝[≠] a] slope f a

full_name: sub_smul_dslope
code: @[simp]
theorem sub_smul_dslope (f : 𝕜 → E) (a b : 𝕜) : (b - a) • dslope f a b = f b - f a 
goal: (b - a) • dslope f a b = f b - f a

full_name: dslope_sub_smul_of_ne
code: theorem dslope_sub_smul_of_ne (f : 𝕜 → E) (h : b ≠ a) :
    dslope (fun x => (x - a) • f x) a b = f b 
goal: dslope (fun x => (x - a) • f x) a b = f b

full_name: eqOn_dslope_sub_smul
code: theorem eqOn_dslope_sub_smul (f : 𝕜 → E) (a : 𝕜) :
    EqOn (dslope (fun x => (x - a) • f x) a) f {a}ᶜ 
goal: Set.EqOn (dslope (fun x => (x - a) • f x) a) f {a}ᶜ

full_name: dslope_sub_smul
code: theorem dslope_sub_smul [DecidableEq 𝕜] (f : 𝕜 → E) (a : 𝕜) :
    dslope (fun x => (x - a) • f x) a = update f a (deriv (fun x => (x - a) • f x) a) 
goal: dslope (fun x => (x - a) • f x) a = Function.update f a (deriv (fun x => (x - a) • f x) a)

full_name: continuousAt_dslope_same
code: @[simp]
theorem continuousAt_dslope_same : ContinuousAt (dslope f a) a ↔ DifferentiableAt 𝕜 f a 
goal: ContinuousAt (dslope f a) a ↔ DifferentiableAt 𝕜 f a

full_name: ContinuousWithinAt.of_dslope
code: theorem ContinuousWithinAt.of_dslope (h : ContinuousWithinAt (dslope f a) s b) :
    ContinuousWithinAt f s b 
goal: ContinuousWithinAt f s b

full_name: ContinuousAt.of_dslope
code: theorem ContinuousAt.of_dslope (h : ContinuousAt (dslope f a) b) : ContinuousAt f b 
goal: ContinuousAt f b

full_name: ContinuousOn.of_dslope
code: theorem ContinuousOn.of_dslope (h : ContinuousOn (dslope f a) s) : ContinuousOn f s 
goal: ContinuousOn f s

full_name: continuousWithinAt_dslope_of_ne
code: theorem continuousWithinAt_dslope_of_ne (h : b ≠ a) :
    ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b 
goal: ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b

full_name: continuousAt_dslope_of_ne
code: theorem continuousAt_dslope_of_ne (h : b ≠ a) : ContinuousAt (dslope f a) b ↔ ContinuousAt f b 
goal: ContinuousAt (dslope f a) b ↔ ContinuousAt f b

full_name: continuousOn_dslope
code: theorem continuousOn_dslope (h : s ∈ 𝓝 a) :
    ContinuousOn (dslope f a) s ↔ ContinuousOn f s ∧ DifferentiableAt 𝕜 f a 
goal: ContinuousOn (dslope f a) s ↔ ContinuousOn f s ∧ DifferentiableAt 𝕜 f a

full_name: DifferentiableWithinAt.of_dslope
code: theorem DifferentiableWithinAt.of_dslope (h : DifferentiableWithinAt 𝕜 (dslope f a) s b) :
    DifferentiableWithinAt 𝕜 f s b 
goal: DifferentiableWithinAt 𝕜 f s b

full_name: DifferentiableAt.of_dslope
code: theorem DifferentiableAt.of_dslope (h : DifferentiableAt 𝕜 (dslope f a) b) :
    DifferentiableAt 𝕜 f b 
goal: DifferentiableAt 𝕜 f b

full_name: DifferentiableOn.of_dslope
code: theorem DifferentiableOn.of_dslope (h : DifferentiableOn 𝕜 (dslope f a) s) :
    DifferentiableOn 𝕜 f s 
goal: DifferentiableOn 𝕜 f s

full_name: differentiableWithinAt_dslope_of_ne
code: theorem differentiableWithinAt_dslope_of_ne (h : b ≠ a) :
    DifferentiableWithinAt 𝕜 (dslope f a) s b ↔ DifferentiableWithinAt 𝕜 f s b 
goal: DifferentiableWithinAt 𝕜 (dslope f a) s b ↔ DifferentiableWithinAt 𝕜 f s b

full_name: differentiableOn_dslope_of_nmem
code: theorem differentiableOn_dslope_of_nmem (h : a ∉ s) :
    DifferentiableOn 𝕜 (dslope f a) s ↔ DifferentiableOn 𝕜 f s 
goal: DifferentiableOn 𝕜 (dslope f a) s ↔ DifferentiableOn 𝕜 f s

full_name: differentiableAt_dslope_of_ne
code: theorem differentiableAt_dslope_of_ne (h : b ≠ a) :
    DifferentiableAt 𝕜 (dslope f a) b ↔ DifferentiableAt 𝕜 f b 
goal: DifferentiableAt 𝕜 (dslope f a) b ↔ DifferentiableAt 𝕜 f b