Documentation

Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Differentiability of trigonometric functions #

Main statements #

The differentiability of the usual trigonometric functions is proved, and their derivatives are computed.

Tags #

sin, cos, tan, angle

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theorem Complex.hasStrictDerivAt_sin (x : ℂ) :
HasStrictDerivAt sin (cos x) x

The complex sine function is everywhere strictly differentiable, with the derivative cos x.

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theorem Complex.hasDerivAt_sin (x : ℂ) :
HasDerivAt sin (cos x) x

The complex sine function is everywhere differentiable, with the derivative cos x.

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theorem Complex.contDiff_sin {n : WithTop ℕ∞} :
ContDiff ℂ n sin
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@[simp]
theorem Complex.differentiable_sin :
Differentiable ℂ sin
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@[simp]
theorem Complex.differentiableAt_sin {x : ℂ} :
DifferentiableAt ℂ sin x
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theorem Complex.analyticAt_sin {x : ℂ} :
AnalyticAt ℂ sin x

The function Complex.sin is complex analytic.

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theorem Complex.analyticWithinAt_sin {x : ℂ} {s : Set ℂ} :
AnalyticWithinAt ℂ sin s x

The function Complex.sin is complex analytic.

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theorem Complex.analyticOnNhd_sin {s : Set ℂ} :
AnalyticOnNhd ℂ sin s

The function Complex.sin is complex analytic.

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theorem Complex.analyticOn_sin {s : Set ℂ} :
AnalyticOn ℂ sin s

The function Complex.sin is complex analytic.

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@[simp]
theorem Complex.deriv_sin :
deriv sin = cos
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theorem Complex.hasStrictDerivAt_cos (x : ℂ) :
HasStrictDerivAt cos (-sin x) x

The complex cosine function is everywhere strictly differentiable, with the derivative -sin x.

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theorem Complex.hasDerivAt_cos (x : ℂ) :
HasDerivAt cos (-sin x) x

The complex cosine function is everywhere differentiable, with the derivative -sin x.

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theorem Complex.contDiff_cos {n : WithTop ℕ∞} :
ContDiff ℂ n cos
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@[simp]
theorem Complex.differentiable_cos :
Differentiable ℂ cos
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@[simp]
theorem Complex.differentiableAt_cos {x : ℂ} :
DifferentiableAt ℂ cos x
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theorem Complex.analyticAt_cos {x : ℂ} :
AnalyticAt ℂ cos x

The function Complex.cos is complex analytic.

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theorem Complex.analyticWithinAt_cos {x : ℂ} {s : Set ℂ} :
AnalyticWithinAt ℂ cos s x

The function Complex.cos is complex analytic.

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theorem Complex.analyticOnNhd_cos {s : Set ℂ} :
AnalyticOnNhd ℂ cos s

The function Complex.cos is complex analytic.

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theorem Complex.analyticOn_cos {s : Set ℂ} :
AnalyticOn ℂ cos s

The function Complex.cos is complex analytic.

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theorem Complex.deriv_cos {x : ℂ} :
deriv cos x = -sin x
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@[simp]
theorem Complex.deriv_cos' :
deriv cos = fun (x : ℂ) => -sin x
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theorem Complex.hasStrictDerivAt_sinh (x : ℂ) :
HasStrictDerivAt sinh (cosh x) x

The complex hyperbolic sine function is everywhere strictly differentiable, with the derivative cosh x.

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theorem Complex.hasDerivAt_sinh (x : ℂ) :
HasDerivAt sinh (cosh x) x

The complex hyperbolic sine function is everywhere differentiable, with the derivative cosh x.

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theorem Complex.contDiff_sinh {n : WithTop ℕ∞} :
ContDiff ℂ n sinh
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@[simp]
theorem Complex.differentiable_sinh :
Differentiable ℂ sinh
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@[simp]
theorem Complex.differentiableAt_sinh {x : ℂ} :
DifferentiableAt ℂ sinh x
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theorem Complex.analyticAt_sinh {x : ℂ} :
AnalyticAt ℂ sinh x

The function Complex.sinh is complex analytic.

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theorem Complex.analyticWithinAt_sinh {x : ℂ} {s : Set ℂ} :
AnalyticWithinAt ℂ sinh s x

The function Complex.sinh is complex analytic.

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theorem Complex.analyticOnNhd_sinh {s : Set ℂ} :
AnalyticOnNhd ℂ sinh s

The function Complex.sinh is complex analytic.

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theorem Complex.analyticOn_sinh {s : Set ℂ} :
AnalyticOn ℂ sinh s

The function Complex.sinh is complex analytic.

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@[simp]
theorem Complex.deriv_sinh :
deriv sinh = cosh
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theorem Complex.hasStrictDerivAt_cosh (x : ℂ) :
HasStrictDerivAt cosh (sinh x) x

The complex hyperbolic cosine function is everywhere strictly differentiable, with the derivative sinh x.

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theorem Complex.hasDerivAt_cosh (x : ℂ) :
HasDerivAt cosh (sinh x) x

The complex hyperbolic cosine function is everywhere differentiable, with the derivative sinh x.

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theorem Complex.contDiff_cosh {n : WithTop ℕ∞} :
ContDiff ℂ n cosh
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@[simp]
theorem Complex.differentiable_cosh :
Differentiable ℂ cosh
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@[simp]
theorem Complex.differentiableAt_cosh {x : ℂ} :
DifferentiableAt ℂ cosh x
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theorem Complex.analyticAt_cosh {x : ℂ} :
AnalyticAt ℂ cosh x

The function Complex.cosh is complex analytic.

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theorem Complex.analyticWithinAt_cosh {x : ℂ} {s : Set ℂ} :
AnalyticWithinAt ℂ cosh s x

The function Complex.cosh is complex analytic.

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theorem Complex.analyticOnNhd_cosh {s : Set ℂ} :
AnalyticOnNhd ℂ cosh s

The function Complex.cosh is complex analytic.

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theorem Complex.analyticOn_cosh {s : Set ℂ} :
AnalyticOn ℂ cosh s

The function Complex.cosh is complex analytic.

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@[simp]
theorem Complex.deriv_cosh :
deriv cosh = sinh

Simp lemmas for derivatives of fun x => Complex.cos (f x) etc., f : ℂ → ℂ #

Complex.cos #

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theorem HasStrictDerivAt.ccos {f : ℂ → ℂ} {f' x : ℂ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℂ) => Complex.cos (f x)) (-Complex.sin (f x) * f') x
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theorem HasDerivAt.ccos {f : ℂ → ℂ} {f' x : ℂ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℂ) => Complex.cos (f x)) (-Complex.sin (f x) * f') x
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theorem HasDerivWithinAt.ccos {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℂ) => Complex.cos (f x)) (-Complex.sin (f x) * f') s x
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theorem derivWithin_ccos {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
derivWithin (fun (x : ℂ) => Complex.cos (f x)) s x = -Complex.sin (f x) * derivWithin f s x
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@[simp]
theorem deriv_ccos {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) :
deriv (fun (x : ℂ) => Complex.cos (f x)) x = -Complex.sin (f x) * deriv f x

Complex.sin #

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theorem HasStrictDerivAt.csin {f : ℂ → ℂ} {f' x : ℂ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℂ) => Complex.sin (f x)) (Complex.cos (f x) * f') x
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theorem HasDerivAt.csin {f : ℂ → ℂ} {f' x : ℂ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℂ) => Complex.sin (f x)) (Complex.cos (f x) * f') x
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theorem HasDerivWithinAt.csin {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℂ) => Complex.sin (f x)) (Complex.cos (f x) * f') s x
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theorem derivWithin_csin {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
derivWithin (fun (x : ℂ) => Complex.sin (f x)) s x = Complex.cos (f x) * derivWithin f s x
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@[simp]
theorem deriv_csin {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) :
deriv (fun (x : ℂ) => Complex.sin (f x)) x = Complex.cos (f x) * deriv f x

Complex.cosh #

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theorem HasStrictDerivAt.ccosh {f : ℂ → ℂ} {f' x : ℂ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℂ) => Complex.cosh (f x)) (Complex.sinh (f x) * f') x
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theorem HasDerivAt.ccosh {f : ℂ → ℂ} {f' x : ℂ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℂ) => Complex.cosh (f x)) (Complex.sinh (f x) * f') x
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theorem HasDerivWithinAt.ccosh {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℂ) => Complex.cosh (f x)) (Complex.sinh (f x) * f') s x
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theorem derivWithin_ccosh {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
derivWithin (fun (x : ℂ) => Complex.cosh (f x)) s x = Complex.sinh (f x) * derivWithin f s x
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@[simp]
theorem deriv_ccosh {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) :
deriv (fun (x : ℂ) => Complex.cosh (f x)) x = Complex.sinh (f x) * deriv f x

Complex.sinh #

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theorem HasStrictDerivAt.csinh {f : ℂ → ℂ} {f' x : ℂ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℂ) => Complex.sinh (f x)) (Complex.cosh (f x) * f') x
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theorem HasDerivAt.csinh {f : ℂ → ℂ} {f' x : ℂ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℂ) => Complex.sinh (f x)) (Complex.cosh (f x) * f') x
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theorem HasDerivWithinAt.csinh {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℂ) => Complex.sinh (f x)) (Complex.cosh (f x) * f') s x
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theorem derivWithin_csinh {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
derivWithin (fun (x : ℂ) => Complex.sinh (f x)) s x = Complex.cosh (f x) * derivWithin f s x
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@[simp]
theorem deriv_csinh {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) :
deriv (fun (x : ℂ) => Complex.sinh (f x)) x = Complex.cosh (f x) * deriv f x

Simp lemmas for derivatives of fun x => Complex.cos (f x) etc., f : E → ℂ #

Complex.cos #

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theorem HasStrictFDerivAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Complex.cos (f x)) (-Complex.sin (f x) • f') x
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theorem HasFDerivAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Complex.cos (f x)) (-Complex.sin (f x) • f') x
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theorem HasFDerivWithinAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Complex.cos (f x)) (-Complex.sin (f x) • f') s x
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theorem DifferentiableWithinAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) :
DifferentiableWithinAt ℂ (fun (x : E) => Complex.cos (f x)) s x
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@[simp]
theorem DifferentiableAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
DifferentiableAt ℂ (fun (x : E) => Complex.cos (f x)) x
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theorem DifferentiableOn.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s) :
DifferentiableOn ℂ (fun (x : E) => Complex.cos (f x)) s
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@[simp]
theorem Differentiable.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) :
Differentiable ℂ fun (x : E) => Complex.cos (f x)
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theorem fderivWithin_ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
fderivWithin ℂ (fun (x : E) => Complex.cos (f x)) s x = -Complex.sin (f x) • fderivWithin ℂ f s x
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@[simp]
theorem fderiv_ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
fderiv ℂ (fun (x : E) => Complex.cos (f x)) x = -Complex.sin (f x) • fderiv ℂ f x
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theorem ContDiff.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : WithTop ℕ∞} (h : ContDiff ℂ n f) :
ContDiff ℂ n fun (x : E) => Complex.cos (f x)
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theorem ContDiffAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℂ n f x) :
ContDiffAt ℂ n (fun (x : E) => Complex.cos (f x)) x
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theorem ContDiffOn.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℂ n f s) :
ContDiffOn ℂ n (fun (x : E) => Complex.cos (f x)) s
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theorem ContDiffWithinAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) :
ContDiffWithinAt ℂ n (fun (x : E) => Complex.cos (f x)) s x

Complex.sin #

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theorem HasStrictFDerivAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Complex.sin (f x)) (Complex.cos (f x) • f') x
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theorem HasFDerivAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Complex.sin (f x)) (Complex.cos (f x) • f') x
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theorem HasFDerivWithinAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Complex.sin (f x)) (Complex.cos (f x) • f') s x
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theorem DifferentiableWithinAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) :
DifferentiableWithinAt ℂ (fun (x : E) => Complex.sin (f x)) s x
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@[simp]
theorem DifferentiableAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
DifferentiableAt ℂ (fun (x : E) => Complex.sin (f x)) x
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theorem DifferentiableOn.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s) :
DifferentiableOn ℂ (fun (x : E) => Complex.sin (f x)) s
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@[simp]
theorem Differentiable.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) :
Differentiable ℂ fun (x : E) => Complex.sin (f x)
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theorem fderivWithin_csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
fderivWithin ℂ (fun (x : E) => Complex.sin (f x)) s x = Complex.cos (f x) • fderivWithin ℂ f s x
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@[simp]
theorem fderiv_csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
fderiv ℂ (fun (x : E) => Complex.sin (f x)) x = Complex.cos (f x) • fderiv ℂ f x
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theorem ContDiff.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : WithTop ℕ∞} (h : ContDiff ℂ n f) :
ContDiff ℂ n fun (x : E) => Complex.sin (f x)
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theorem ContDiffAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℂ n f x) :
ContDiffAt ℂ n (fun (x : E) => Complex.sin (f x)) x
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theorem ContDiffOn.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℂ n f s) :
ContDiffOn ℂ n (fun (x : E) => Complex.sin (f x)) s
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theorem ContDiffWithinAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) :
ContDiffWithinAt ℂ n (fun (x : E) => Complex.sin (f x)) s x

Complex.cosh #

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theorem HasStrictFDerivAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Complex.cosh (f x)) (Complex.sinh (f x) • f') x
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theorem HasFDerivAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Complex.cosh (f x)) (Complex.sinh (f x) • f') x
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theorem HasFDerivWithinAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Complex.cosh (f x)) (Complex.sinh (f x) • f') s x
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theorem DifferentiableWithinAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) :
DifferentiableWithinAt ℂ (fun (x : E) => Complex.cosh (f x)) s x
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@[simp]
theorem DifferentiableAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
DifferentiableAt ℂ (fun (x : E) => Complex.cosh (f x)) x
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theorem DifferentiableOn.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s) :
DifferentiableOn ℂ (fun (x : E) => Complex.cosh (f x)) s
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@[simp]
theorem Differentiable.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) :
Differentiable ℂ fun (x : E) => Complex.cosh (f x)
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theorem fderivWithin_ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
fderivWithin ℂ (fun (x : E) => Complex.cosh (f x)) s x = Complex.sinh (f x) • fderivWithin ℂ f s x
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@[simp]
theorem fderiv_ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
fderiv ℂ (fun (x : E) => Complex.cosh (f x)) x = Complex.sinh (f x) • fderiv ℂ f x
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theorem ContDiff.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : WithTop ℕ∞} (h : ContDiff ℂ n f) :
ContDiff ℂ n fun (x : E) => Complex.cosh (f x)
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theorem ContDiffAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℂ n f x) :
ContDiffAt ℂ n (fun (x : E) => Complex.cosh (f x)) x
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theorem ContDiffOn.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℂ n f s) :
ContDiffOn ℂ n (fun (x : E) => Complex.cosh (f x)) s
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theorem ContDiffWithinAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) :
ContDiffWithinAt ℂ n (fun (x : E) => Complex.cosh (f x)) s x

Complex.sinh #

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theorem HasStrictFDerivAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Complex.sinh (f x)) (Complex.cosh (f x) • f') x
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theorem HasFDerivAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Complex.sinh (f x)) (Complex.cosh (f x) • f') x
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theorem HasFDerivWithinAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Complex.sinh (f x)) (Complex.cosh (f x) • f') s x
source
theorem DifferentiableWithinAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) :
DifferentiableWithinAt ℂ (fun (x : E) => Complex.sinh (f x)) s x
source
@[simp]
theorem DifferentiableAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
DifferentiableAt ℂ (fun (x : E) => Complex.sinh (f x)) x
source
theorem DifferentiableOn.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s) :
DifferentiableOn ℂ (fun (x : E) => Complex.sinh (f x)) s
source
@[simp]
theorem Differentiable.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) :
Differentiable ℂ fun (x : E) => Complex.sinh (f x)
source
theorem fderivWithin_csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x) :
fderivWithin ℂ (fun (x : E) => Complex.sinh (f x)) s x = Complex.cosh (f x) • fderivWithin ℂ f s x
source
@[simp]
theorem fderiv_csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x) :
fderiv ℂ (fun (x : E) => Complex.sinh (f x)) x = Complex.cosh (f x) • fderiv ℂ f x
source
theorem ContDiff.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : WithTop ℕ∞} (h : ContDiff ℂ n f) :
ContDiff ℂ n fun (x : E) => Complex.sinh (f x)
source
theorem ContDiffAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℂ n f x) :
ContDiffAt ℂ n (fun (x : E) => Complex.sinh (f x)) x
source
theorem ContDiffOn.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℂ n f s) :
ContDiffOn ℂ n (fun (x : E) => Complex.sinh (f x)) s
source
theorem ContDiffWithinAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) :
ContDiffWithinAt ℂ n (fun (x : E) => Complex.sinh (f x)) s x
source
theorem Real.hasStrictDerivAt_sin (x : ℝ) :
HasStrictDerivAt sin (cos x) x
source
theorem Real.hasDerivAt_sin (x : ℝ) :
HasDerivAt sin (cos x) x
source
theorem Real.contDiff_sin {n : WithTop ℕ∞} :
ContDiff ℝ n sin
source
@[simp]
theorem Real.differentiable_sin :
Differentiable ℝ sin
source
@[simp]
theorem Real.differentiableAt_sin {x : ℝ} :
DifferentiableAt ℝ sin x
source
theorem Real.analyticAt_sin {x : ℝ} :
AnalyticAt ℝ sin x

The function Real.sin is real analytic.

source
theorem Real.analyticWithinAt_sin {x : ℝ} {s : Set ℝ} :
AnalyticWithinAt ℝ sin s x

The function Real.sin is real analytic.

source
theorem Real.analyticOnNhd_sin {s : Set ℝ} :
AnalyticOnNhd ℝ sin s

The function Real.sin is real analytic.

source
theorem Real.analyticOn_sin {s : Set ℝ} :
AnalyticOn ℝ sin s

The function Real.sin is real analytic.

source
@[simp]
theorem Real.deriv_sin :
deriv sin = cos
source
theorem Real.hasStrictDerivAt_cos (x : ℝ) :
HasStrictDerivAt cos (-sin x) x
source
theorem Real.hasDerivAt_cos (x : ℝ) :
HasDerivAt cos (-sin x) x
source
theorem Real.contDiff_cos {n : WithTop ℕ∞} :
ContDiff ℝ n cos
source
@[simp]
theorem Real.differentiable_cos :
Differentiable ℝ cos
source
@[simp]
theorem Real.differentiableAt_cos {x : ℝ} :
DifferentiableAt ℝ cos x
source
theorem Real.analyticAt_cos {x : ℝ} :
AnalyticAt ℝ cos x

The function Real.cos is real analytic.

source
theorem Real.analyticWithinAt_cos {x : ℝ} {s : Set ℝ} :
AnalyticWithinAt ℝ cos s x

The function Real.cos is real analytic.

source
theorem Real.analyticOnNhd_cos {s : Set ℝ} :
AnalyticOnNhd ℝ cos s

The function Real.cos is real analytic.

source
theorem Real.analyticOn_cos {s : Set ℝ} :
AnalyticOn ℝ cos s

The function Real.cos is real analytic.

source
theorem Real.deriv_cos {x : ℝ} :
deriv cos x = -sin x
source
@[simp]
theorem Real.deriv_cos' :
deriv cos = fun (x : ℝ) => -sin x
source
theorem Real.hasStrictDerivAt_sinh (x : ℝ) :
HasStrictDerivAt sinh (cosh x) x
source
theorem Real.hasDerivAt_sinh (x : ℝ) :
HasDerivAt sinh (cosh x) x
source
theorem Real.contDiff_sinh {n : WithTop ℕ∞} :
ContDiff ℝ n sinh
source
@[simp]
theorem Real.differentiable_sinh :
Differentiable ℝ sinh
source
@[simp]
theorem Real.differentiableAt_sinh {x : ℝ} :
DifferentiableAt ℝ sinh x
source
theorem Real.analyticAt_sinh {x : ℝ} :
AnalyticAt ℝ sinh x

The function Real.sinh is real analytic.

source
theorem Real.analyticWithinAt_sinh {x : ℝ} {s : Set ℝ} :
AnalyticWithinAt ℝ sinh s x

The function Real.sinh is real analytic.

source
theorem Real.analyticOnNhd_sinh {s : Set ℝ} :
AnalyticOnNhd ℝ sinh s

The function Real.sinh is real analytic.

source
theorem Real.analyticOn_sinh {s : Set ℝ} :
AnalyticOn ℝ sinh s

The function Real.sinh is real analytic.

source
@[simp]
theorem Real.deriv_sinh :
deriv sinh = cosh
source
theorem Real.hasStrictDerivAt_cosh (x : ℝ) :
HasStrictDerivAt cosh (sinh x) x
source
theorem Real.hasDerivAt_cosh (x : ℝ) :
HasDerivAt cosh (sinh x) x
source
theorem Real.contDiff_cosh {n : WithTop ℕ∞} :
ContDiff ℝ n cosh
source
@[simp]
theorem Real.differentiable_cosh :
Differentiable ℝ cosh
source
@[simp]
theorem Real.differentiableAt_cosh {x : ℝ} :
DifferentiableAt ℝ cosh x
source
theorem Real.analyticAt_cosh {x : ℝ} :
AnalyticAt ℝ cosh x

The function Real.cosh is real analytic.

source
theorem Real.analyticWithinAt_cosh {x : ℝ} {s : Set ℝ} :
AnalyticWithinAt ℝ cosh s x

The function Real.cosh is real analytic.

source
theorem Real.analyticOnNhd_cosh {s : Set ℝ} :
AnalyticOnNhd ℝ cosh s

The function Real.cosh is real analytic.

source
theorem Real.analyticOn_cosh {s : Set ℝ} :
AnalyticOn ℝ cosh s

The function Real.cosh is real analytic.

source
@[simp]
theorem Real.deriv_cosh :
deriv cosh = sinh
source
theorem Real.sinh_strictMono :
StrictMono sinh

sinh is strictly monotone.

source
theorem Real.sinh_injective :
Function.Injective sinh

sinh is injective, ∀ a b, sinh a = sinh b → a = b.

source
@[simp]
theorem Real.sinh_inj {x y : ℝ} :
sinh x = sinh y ↔ x = y
source
@[simp]
theorem Real.sinh_le_sinh {x y : ℝ} :
sinh x ≤ sinh y ↔ x ≤ y
source
@[simp]
theorem Real.sinh_lt_sinh {x y : ℝ} :
sinh x < sinh y ↔ x < y
source
@[simp]
theorem Real.sinh_eq_zero {x : ℝ} :
sinh x = 0 ↔ x = 0
source
theorem Real.sinh_ne_zero {x : ℝ} :
sinh x ≠ 0 ↔ x ≠ 0
source
@[simp]
theorem Real.sinh_pos_iff {x : ℝ} :
0 < sinh x ↔ 0 < x
source
@[simp]
theorem Real.sinh_nonpos_iff {x : ℝ} :
sinh x ≤ 0 ↔ x ≤ 0
source
@[simp]
theorem Real.sinh_neg_iff {x : ℝ} :
sinh x < 0 ↔ x < 0
source
@[simp]
theorem Real.sinh_nonneg_iff {x : ℝ} :
0 ≤ sinh x ↔ 0 ≤ x
source
theorem Real.abs_sinh (x : ℝ) :
|sinh x| = sinh |x|
source
theorem Real.cosh_strictMonoOn :
StrictMonoOn cosh (Set.Ici 0)
source
@[simp]
theorem Real.cosh_le_cosh {x y : ℝ} :
cosh x ≤ cosh y ↔ |x| ≤ |y|
source
@[simp]
theorem Real.cosh_lt_cosh {x y : ℝ} :
cosh x < cosh y ↔ |x| < |y|
source
@[simp]
theorem Real.one_le_cosh (x : ℝ) :
1 ≤ cosh x
source
@[simp]
theorem Real.one_lt_cosh {x : ℝ} :
1 < cosh x ↔ x ≠ 0
source
theorem Real.sinh_sub_id_strictMono :
StrictMono fun (x : ℝ) => sinh x - x
source
@[simp]
theorem Real.self_le_sinh_iff {x : ℝ} :
x ≤ sinh x ↔ 0 ≤ x
source
@[simp]
theorem Real.sinh_le_self_iff {x : ℝ} :
sinh x ≤ x ↔ x ≤ 0
source
@[simp]
theorem Real.self_lt_sinh_iff {x : ℝ} :
x < sinh x ↔ 0 < x
source
@[simp]
theorem Real.sinh_lt_self_iff {x : ℝ} :
sinh x < x ↔ x < 0

Simp lemmas for derivatives of fun x => Real.cos (f x) etc., f : ℝ → ℝ #

Real.cos #

source
theorem HasStrictDerivAt.cos {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℝ) => Real.cos (f x)) (-Real.sin (f x) * f') x
source
theorem HasDerivAt.cos {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℝ) => Real.cos (f x)) (-Real.sin (f x) * f') x
source
theorem HasDerivWithinAt.cos {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℝ) => Real.cos (f x)) (-Real.sin (f x) * f') s x
source
theorem derivWithin_cos {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
derivWithin (fun (x : ℝ) => Real.cos (f x)) s x = -Real.sin (f x) * derivWithin f s x
source
@[simp]
theorem deriv_cos {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) :
deriv (fun (x : ℝ) => Real.cos (f x)) x = -Real.sin (f x) * deriv f x

Real.sin #

source
theorem HasStrictDerivAt.sin {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℝ) => Real.sin (f x)) (Real.cos (f x) * f') x
source
theorem HasDerivAt.sin {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℝ) => Real.sin (f x)) (Real.cos (f x) * f') x
source
theorem HasDerivWithinAt.sin {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℝ) => Real.sin (f x)) (Real.cos (f x) * f') s x
source
theorem derivWithin_sin {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
derivWithin (fun (x : ℝ) => Real.sin (f x)) s x = Real.cos (f x) * derivWithin f s x
source
@[simp]
theorem deriv_sin {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) :
deriv (fun (x : ℝ) => Real.sin (f x)) x = Real.cos (f x) * deriv f x

Real.cosh #

source
theorem HasStrictDerivAt.cosh {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℝ) => Real.cosh (f x)) (Real.sinh (f x) * f') x
source
theorem HasDerivAt.cosh {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℝ) => Real.cosh (f x)) (Real.sinh (f x) * f') x
source
theorem HasDerivWithinAt.cosh {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℝ) => Real.cosh (f x)) (Real.sinh (f x) * f') s x
source
theorem derivWithin_cosh {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
derivWithin (fun (x : ℝ) => Real.cosh (f x)) s x = Real.sinh (f x) * derivWithin f s x
source
@[simp]
theorem deriv_cosh {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) :
deriv (fun (x : ℝ) => Real.cosh (f x)) x = Real.sinh (f x) * deriv f x

Real.sinh #

source
theorem HasStrictDerivAt.sinh {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x) :
HasStrictDerivAt (fun (x : ℝ) => Real.sinh (f x)) (Real.cosh (f x) * f') x
source
theorem HasDerivAt.sinh {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (fun (x : ℝ) => Real.sinh (f x)) (Real.cosh (f x) * f') x
source
theorem HasDerivWithinAt.sinh {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun (x : ℝ) => Real.sinh (f x)) (Real.cosh (f x) * f') s x
source
theorem derivWithin_sinh {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
derivWithin (fun (x : ℝ) => Real.sinh (f x)) s x = Real.cosh (f x) * derivWithin f s x
source
@[simp]
theorem deriv_sinh {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) :
deriv (fun (x : ℝ) => Real.sinh (f x)) x = Real.cosh (f x) * deriv f x

Simp lemmas for derivatives of fun x => Real.cos (f x) etc., f : E → ℝ #

Real.cos #

source
theorem HasStrictFDerivAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Real.cos (f x)) (-Real.sin (f x) • f') x
source
theorem HasFDerivAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Real.cos (f x)) (-Real.sin (f x) • f') x
source
theorem HasFDerivWithinAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Real.cos (f x)) (-Real.sin (f x) • f') s x
source
theorem DifferentiableWithinAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) :
DifferentiableWithinAt ℝ (fun (x : E) => Real.cos (f x)) s x
source
@[simp]
theorem DifferentiableAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
DifferentiableAt ℝ (fun (x : E) => Real.cos (f x)) x
source
theorem DifferentiableOn.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s) :
DifferentiableOn ℝ (fun (x : E) => Real.cos (f x)) s
source
@[simp]
theorem Differentiable.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) :
Differentiable ℝ fun (x : E) => Real.cos (f x)
source
theorem fderivWithin_cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
fderivWithin ℝ (fun (x : E) => Real.cos (f x)) s x = -Real.sin (f x) • fderivWithin ℝ f s x
source
@[simp]
theorem fderiv_cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
fderiv ℝ (fun (x : E) => Real.cos (f x)) x = -Real.sin (f x) • fderiv ℝ f x
source
theorem ContDiff.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} (h : ContDiff ℝ n f) :
ContDiff ℝ n fun (x : E) => Real.cos (f x)
source
theorem ContDiffAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℝ n f x) :
ContDiffAt ℝ n (fun (x : E) => Real.cos (f x)) x
source
theorem ContDiffOn.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℝ n f s) :
ContDiffOn ℝ n (fun (x : E) => Real.cos (f x)) s
source
theorem ContDiffWithinAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) :
ContDiffWithinAt ℝ n (fun (x : E) => Real.cos (f x)) s x

Real.sin #

source
theorem HasStrictFDerivAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Real.sin (f x)) (Real.cos (f x) • f') x
source
theorem HasFDerivAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Real.sin (f x)) (Real.cos (f x) • f') x
source
theorem HasFDerivWithinAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Real.sin (f x)) (Real.cos (f x) • f') s x
source
theorem DifferentiableWithinAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) :
DifferentiableWithinAt ℝ (fun (x : E) => Real.sin (f x)) s x
source
@[simp]
theorem DifferentiableAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
DifferentiableAt ℝ (fun (x : E) => Real.sin (f x)) x
source
theorem DifferentiableOn.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s) :
DifferentiableOn ℝ (fun (x : E) => Real.sin (f x)) s
source
@[simp]
theorem Differentiable.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) :
Differentiable ℝ fun (x : E) => Real.sin (f x)
source
theorem fderivWithin_sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
fderivWithin ℝ (fun (x : E) => Real.sin (f x)) s x = Real.cos (f x) • fderivWithin ℝ f s x
source
@[simp]
theorem fderiv_sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
fderiv ℝ (fun (x : E) => Real.sin (f x)) x = Real.cos (f x) • fderiv ℝ f x
source
theorem ContDiff.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} (h : ContDiff ℝ n f) :
ContDiff ℝ n fun (x : E) => Real.sin (f x)
source
theorem ContDiffAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℝ n f x) :
ContDiffAt ℝ n (fun (x : E) => Real.sin (f x)) x
source
theorem ContDiffOn.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℝ n f s) :
ContDiffOn ℝ n (fun (x : E) => Real.sin (f x)) s
source
theorem ContDiffWithinAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) :
ContDiffWithinAt ℝ n (fun (x : E) => Real.sin (f x)) s x

Real.cosh #

source
theorem HasStrictFDerivAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Real.cosh (f x)) (Real.sinh (f x) • f') x
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theorem HasFDerivAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Real.cosh (f x)) (Real.sinh (f x) • f') x
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theorem HasFDerivWithinAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Real.cosh (f x)) (Real.sinh (f x) • f') s x
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theorem DifferentiableWithinAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) :
DifferentiableWithinAt ℝ (fun (x : E) => Real.cosh (f x)) s x
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@[simp]
theorem DifferentiableAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
DifferentiableAt ℝ (fun (x : E) => Real.cosh (f x)) x
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theorem DifferentiableOn.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s) :
DifferentiableOn ℝ (fun (x : E) => Real.cosh (f x)) s
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@[simp]
theorem Differentiable.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) :
Differentiable ℝ fun (x : E) => Real.cosh (f x)
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theorem fderivWithin_cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
fderivWithin ℝ (fun (x : E) => Real.cosh (f x)) s x = Real.sinh (f x) • fderivWithin ℝ f s x
source
@[simp]
theorem fderiv_cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
fderiv ℝ (fun (x : E) => Real.cosh (f x)) x = Real.sinh (f x) • fderiv ℝ f x
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theorem ContDiff.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} (h : ContDiff ℝ n f) :
ContDiff ℝ n fun (x : E) => Real.cosh (f x)
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theorem ContDiffAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℝ n f x) :
ContDiffAt ℝ n (fun (x : E) => Real.cosh (f x)) x
source
theorem ContDiffOn.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℝ n f s) :
ContDiffOn ℝ n (fun (x : E) => Real.cosh (f x)) s
source
theorem ContDiffWithinAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) :
ContDiffWithinAt ℝ n (fun (x : E) => Real.cosh (f x)) s x

Real.sinh #

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theorem HasStrictFDerivAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun (x : E) => Real.sinh (f x)) (Real.cosh (f x) • f') x
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theorem HasFDerivAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x) :
HasFDerivAt (fun (x : E) => Real.sinh (f x)) (Real.cosh (f x) • f') x
source
theorem HasFDerivWithinAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (fun (x : E) => Real.sinh (f x)) (Real.cosh (f x) • f') s x
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theorem DifferentiableWithinAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) :
DifferentiableWithinAt ℝ (fun (x : E) => Real.sinh (f x)) s x
source
@[simp]
theorem DifferentiableAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
DifferentiableAt ℝ (fun (x : E) => Real.sinh (f x)) x
source
theorem DifferentiableOn.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s) :
DifferentiableOn ℝ (fun (x : E) => Real.sinh (f x)) s
source
@[simp]
theorem Differentiable.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) :
Differentiable ℝ fun (x : E) => Real.sinh (f x)
source
theorem fderivWithin_sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
fderivWithin ℝ (fun (x : E) => Real.sinh (f x)) s x = Real.cosh (f x) • fderivWithin ℝ f s x
source
@[simp]
theorem fderiv_sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x) :
fderiv ℝ (fun (x : E) => Real.sinh (f x)) x = Real.cosh (f x) • fderiv ℝ f x
source
theorem ContDiff.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} (h : ContDiff ℝ n f) :
ContDiff ℝ n fun (x : E) => Real.sinh (f x)
source
theorem ContDiffAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : WithTop ℕ∞} (hf : ContDiffAt ℝ n f x) :
ContDiffAt ℝ n (fun (x : E) => Real.sinh (f x)) x
source
theorem ContDiffOn.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffOn ℝ n f s) :
ContDiffOn ℝ n (fun (x : E) => Real.sinh (f x)) s
source
theorem ContDiffWithinAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : WithTop ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) :
ContDiffWithinAt ℝ n (fun (x : E) => Real.sinh (f x)) s x
source
@[simp]
theorem Complex.logDeriv_sin :
logDeriv sin = cot
source
@[simp]
theorem Real.logDeriv_sin :
logDeriv sin = cot
source
@[simp]
theorem Complex.logDeriv_cos :
logDeriv cos = -tan
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@[simp]
theorem Real.logDeriv_cos :
logDeriv cos = -tan
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@[simp]
theorem Complex.logDeriv_cosh :
logDeriv cosh = tanh
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@[simp]
theorem Real.logDeriv_cosh :
logDeriv cosh = tanh
source
@[simp]
theorem Complex.LogDeriv_exp :
logDeriv exp = 1
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@[simp]
theorem Real.LogDeriv_exp :
logDeriv exp = 1
source
def Mathlib.Meta.Positivity.evalSinh :
PositivityExt

Extension for the positivity tactic: Real.sinh is positive/nonnegative/nonzero if its input is.

Instances For