Documentation

Mathlib.Analysis.SpecialFunctions.Sqrt

Smoothness of Real.sqrt #

In this file we prove that Real.sqrt is infinitely smooth at all points x ≠ 0 and provide some dot-notation lemmas.

Tags #

sqrt, differentiable

Local homeomorph between (0, +∞) and (0, +∞) with toFun = (· ^ 2) and invFun = Real.sqrt.

Equations
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Instances For
    theorem Real.deriv_sqrt_aux {x : } (hx : x 0) :
    HasStrictDerivAt (fun (x : ) => x) (1 / (2 * x)) x ∀ (n : ℕ∞), ContDiffAt n (fun (x : ) => x) x
    theorem Real.hasStrictDerivAt_sqrt {x : } (hx : x 0) :
    HasStrictDerivAt (fun (x : ) => x) (1 / (2 * x)) x
    theorem Real.contDiffAt_sqrt {x : } {n : ℕ∞} (hx : x 0) :
    ContDiffAt n (fun (x : ) => x) x
    theorem Real.hasDerivAt_sqrt {x : } (hx : x 0) :
    HasDerivAt (fun (x : ) => x) (1 / (2 * x)) x
    theorem HasDerivWithinAt.sqrt {f : } {s : Set } {f' : } {x : } (hf : HasDerivWithinAt f f' s x) (hx : f x 0) :
    HasDerivWithinAt (fun (y : ) => (f y)) (f' / (2 * (f x))) s x
    theorem HasDerivAt.sqrt {f : } {f' : } {x : } (hf : HasDerivAt f f' x) (hx : f x 0) :
    HasDerivAt (fun (y : ) => (f y)) (f' / (2 * (f x))) x
    theorem HasStrictDerivAt.sqrt {f : } {f' : } {x : } (hf : HasStrictDerivAt f f' x) (hx : f x 0) :
    HasStrictDerivAt (fun (t : ) => (f t)) (f' / (2 * (f x))) x
    theorem derivWithin_sqrt {f : } {s : Set } {x : } (hf : DifferentiableWithinAt f s x) (hx : f x 0) (hxs : UniqueDiffWithinAt s x) :
    derivWithin (fun (x : ) => (f x)) s x = derivWithin f s x / (2 * (f x))
    @[simp]
    theorem deriv_sqrt {f : } {x : } (hf : DifferentiableAt f x) (hx : f x 0) :
    deriv (fun (x : ) => (f x)) x = deriv f x / (2 * (f x))
    theorem HasFDerivAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {x : E} {f' : E →L[] } (hf : HasFDerivAt f f' x) (hx : f x 0) :
    HasFDerivAt (fun (y : E) => (f y)) ((1 / (2 * (f x))) f') x
    theorem HasStrictFDerivAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {x : E} {f' : E →L[] } (hf : HasStrictFDerivAt f f' x) (hx : f x 0) :
    HasStrictFDerivAt (fun (y : E) => (f y)) ((1 / (2 * (f x))) f') x
    theorem HasFDerivWithinAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {s : Set E} {x : E} {f' : E →L[] } (hf : HasFDerivWithinAt f f' s x) (hx : f x 0) :
    HasFDerivWithinAt (fun (y : E) => (f y)) ((1 / (2 * (f x))) f') s x
    theorem DifferentiableWithinAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {s : Set E} {x : E} (hf : DifferentiableWithinAt f s x) (hx : f x 0) :
    DifferentiableWithinAt (fun (y : E) => (f y)) s x
    theorem DifferentiableAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {x : E} (hf : DifferentiableAt f x) (hx : f x 0) :
    DifferentiableAt (fun (y : E) => (f y)) x
    theorem DifferentiableOn.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {s : Set E} (hf : DifferentiableOn f s) (hs : xs, f x 0) :
    DifferentiableOn (fun (y : E) => (f y)) s
    theorem Differentiable.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} (hf : Differentiable f) (hs : ∀ (x : E), f x 0) :
    Differentiable fun (y : E) => (f y)
    theorem fderivWithin_sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {s : Set E} {x : E} (hf : DifferentiableWithinAt f s x) (hx : f x 0) (hxs : UniqueDiffWithinAt s x) :
    fderivWithin (fun (x : E) => (f x)) s x = (1 / (2 * (f x))) fderivWithin f s x
    @[simp]
    theorem fderiv_sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {x : E} (hf : DifferentiableAt f x) (hx : f x 0) :
    fderiv (fun (x : E) => (f x)) x = (1 / (2 * (f x))) fderiv f x
    theorem ContDiffAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {n : ℕ∞} {x : E} (hf : ContDiffAt n f x) (hx : f x 0) :
    ContDiffAt n (fun (y : E) => (f y)) x
    theorem ContDiffWithinAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {n : ℕ∞} {s : Set E} {x : E} (hf : ContDiffWithinAt n f s x) (hx : f x 0) :
    ContDiffWithinAt n (fun (y : E) => (f y)) s x
    theorem ContDiffOn.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {n : ℕ∞} {s : Set E} (hf : ContDiffOn n f s) (hs : xs, f x 0) :
    ContDiffOn n (fun (y : E) => (f y)) s
    theorem ContDiff.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {n : ℕ∞} (hf : ContDiff n f) (h : ∀ (x : E), f x 0) :
    ContDiff n fun (y : E) => (f y)