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

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

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

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

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

theorem Complex.contDiff_sin {n : ℕ∞} : ContDiff ℂ n Complex.sin

theorem Complex.differentiable_sin : Differentiable ℂ Complex.sin

theorem Complex.differentiableAt_sin {x : ℂ} : DifferentiableAt ℂ Complex.sin x

@[simp]

theorem Complex.deriv_sin : deriv Complex.sin = Complex.cos

theorem Complex.hasStrictDerivAt_cos (x : ℂ) : HasStrictDerivAt Complex.cos (-Complex.sin x) x

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

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

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

theorem Complex.contDiff_cos {n : ℕ∞} : ContDiff ℂ n Complex.cos

theorem Complex.differentiable_cos : Differentiable ℂ Complex.cos

theorem Complex.differentiableAt_cos {x : ℂ} : DifferentiableAt ℂ Complex.cos x

theorem Complex.deriv_cos {x : ℂ} : deriv Complex.cos x = -Complex.sin x

@[simp]

theorem Complex.deriv_cos' : deriv Complex.cos = fun x => -Complex.sin x

theorem Complex.hasStrictDerivAt_sinh (x : ℂ) : HasStrictDerivAt Complex.sinh (Complex.cosh x) x

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

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

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

theorem Complex.contDiff_sinh {n : ℕ∞} : ContDiff ℂ n Complex.sinh

theorem Complex.differentiable_sinh : Differentiable ℂ Complex.sinh

theorem Complex.differentiableAt_sinh {x : ℂ} : DifferentiableAt ℂ Complex.sinh x

@[simp]

theorem Complex.deriv_sinh : deriv Complex.sinh = Complex.cosh

theorem Complex.hasStrictDerivAt_cosh (x : ℂ) : HasStrictDerivAt Complex.cosh (Complex.sinh x) x

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

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

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

theorem Complex.contDiff_cosh {n : ℕ∞} : ContDiff ℂ n Complex.cosh

theorem Complex.differentiable_cosh : Differentiable ℂ Complex.cosh

theorem Complex.differentiableAt_cosh {x : ℂ} : DifferentiableAt ℂ Complex.cosh x

@[simp]

theorem Complex.deriv_cosh : deriv Complex.cosh = Complex.sinh

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

Complex.cos

theorem HasStrictDerivAt.ccos {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) * f') x

theorem HasDerivAt.ccos {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) * f') x

theorem HasDerivWithinAt.ccos {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) * f') s x

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 => Complex.cos (f x)) s x = -Complex.sin (f x) * derivWithin f s x

@[simp]

theorem deriv_ccos {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) : deriv (fun x => Complex.cos (f x)) x = -Complex.sin (f x) * deriv f x

Complex.sin

theorem HasStrictDerivAt.csin {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Complex.sin (f x)) (Complex.cos (f x) * f') x

theorem HasDerivAt.csin {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Complex.sin (f x)) (Complex.cos (f x) * f') x

theorem HasDerivWithinAt.csin {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Complex.sin (f x)) (Complex.cos (f x) * f') s x

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 => Complex.sin (f x)) s x = Complex.cos (f x) * derivWithin f s x

@[simp]

theorem deriv_csin {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) : deriv (fun x => Complex.sin (f x)) x = Complex.cos (f x) * deriv f x

Complex.cosh

theorem HasStrictDerivAt.ccosh {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') x

theorem HasDerivAt.ccosh {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') x

theorem HasDerivWithinAt.ccosh {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') s x

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 => Complex.cosh (f x)) s x = Complex.sinh (f x) * derivWithin f s x

@[simp]

theorem deriv_ccosh {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) : deriv (fun x => Complex.cosh (f x)) x = Complex.sinh (f x) * deriv f x

Complex.sinh

theorem HasStrictDerivAt.csinh {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') x

theorem HasDerivAt.csinh {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') x

theorem HasDerivWithinAt.csinh {f : ℂ → ℂ} {f' : ℂ} {x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') s x

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 => Complex.sinh (f x)) s x = Complex.cosh (f x) * derivWithin f s x

@[simp]

theorem deriv_csinh {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x) : deriv (fun x => 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

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 => Complex.cos (f x)) (-Complex.sin (f x) • f') x

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 => Complex.cos (f x)) (-Complex.sin (f x) • f') x

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 => Complex.cos (f x)) (-Complex.sin (f x) • f') s x

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 => Complex.cos (f x)) s x

@[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 => Complex.cos (f x)) x

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 => Complex.cos (f x)) s

@[simp]

theorem Differentiable.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) : Differentiable ℂ fun x => Complex.cos (f x)

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 => Complex.cos (f x)) s x = -Complex.sin (f x) • fderivWithin ℂ f s x

@[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 => Complex.cos (f x)) x = -Complex.sin (f x) • fderiv ℂ f x

theorem ContDiff.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.cos (f x)

theorem ContDiffAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x) : ContDiffAt ℂ n (fun x => Complex.cos (f x)) x

theorem ContDiffOn.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s) : ContDiffOn ℂ n (fun x => Complex.cos (f x)) s

theorem ContDiffWithinAt.ccos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) : ContDiffWithinAt ℂ n (fun x => Complex.cos (f x)) s x

Complex.sin

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 => Complex.sin (f x)) (Complex.cos (f x) • f') x

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 => Complex.sin (f x)) (Complex.cos (f x) • f') x

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 => Complex.sin (f x)) (Complex.cos (f x) • f') s x

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 => Complex.sin (f x)) s x

@[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 => Complex.sin (f x)) x

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 => Complex.sin (f x)) s

@[simp]

theorem Differentiable.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) : Differentiable ℂ fun x => Complex.sin (f x)

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 => Complex.sin (f x)) s x = Complex.cos (f x) • fderivWithin ℂ f s x

@[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 => Complex.sin (f x)) x = Complex.cos (f x) • fderiv ℂ f x

theorem ContDiff.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.sin (f x)

theorem ContDiffAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x) : ContDiffAt ℂ n (fun x => Complex.sin (f x)) x

theorem ContDiffOn.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s) : ContDiffOn ℂ n (fun x => Complex.sin (f x)) s

theorem ContDiffWithinAt.csin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) : ContDiffWithinAt ℂ n (fun x => Complex.sin (f x)) s x

Complex.cosh

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 => Complex.cosh (f x)) (Complex.sinh (f x) • f') x

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 => Complex.cosh (f x)) (Complex.sinh (f x) • f') x

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 => Complex.cosh (f x)) (Complex.sinh (f x) • f') s x

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 => Complex.cosh (f x)) s x

@[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 => Complex.cosh (f x)) x

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 => Complex.cosh (f x)) s

@[simp]

theorem Differentiable.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) : Differentiable ℂ fun x => Complex.cosh (f x)

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 => Complex.cosh (f x)) s x = Complex.sinh (f x) • fderivWithin ℂ f s x

@[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 => Complex.cosh (f x)) x = Complex.sinh (f x) • fderiv ℂ f x

theorem ContDiff.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.cosh (f x)

theorem ContDiffAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x) : ContDiffAt ℂ n (fun x => Complex.cosh (f x)) x

theorem ContDiffOn.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s) : ContDiffOn ℂ n (fun x => Complex.cosh (f x)) s

theorem ContDiffWithinAt.ccosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) : ContDiffWithinAt ℂ n (fun x => Complex.cosh (f x)) s x

Complex.sinh

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 => Complex.sinh (f x)) (Complex.cosh (f x) • f') x

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 => Complex.sinh (f x)) (Complex.cosh (f x) • f') x

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 => Complex.sinh (f x)) (Complex.cosh (f x) • f') s x

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 => Complex.sinh (f x)) s x

@[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 => Complex.sinh (f x)) x

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 => Complex.sinh (f x)) s

@[simp]

theorem Differentiable.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f) : Differentiable ℂ fun x => Complex.sinh (f x)

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 => Complex.sinh (f x)) s x = Complex.cosh (f x) • fderivWithin ℂ f s x

@[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 => Complex.sinh (f x)) x = Complex.cosh (f x) • fderiv ℂ f x

theorem ContDiff.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.sinh (f x)

theorem ContDiffAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x) : ContDiffAt ℂ n (fun x => Complex.sinh (f x)) x

theorem ContDiffOn.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s) : ContDiffOn ℂ n (fun x => Complex.sinh (f x)) s

theorem ContDiffWithinAt.csinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℂ n f s x) : ContDiffWithinAt ℂ n (fun x => Complex.sinh (f x)) s x

theorem Real.hasStrictDerivAt_sin (x : ℝ) : HasStrictDerivAt Real.sin (Real.cos x) x

theorem Real.hasDerivAt_sin (x : ℝ) : HasDerivAt Real.sin (Real.cos x) x

theorem Real.contDiff_sin {n : ℕ∞} : ContDiff ℝ n Real.sin

theorem Real.differentiable_sin : Differentiable ℝ Real.sin

theorem Real.differentiableAt_sin {x : ℝ} : DifferentiableAt ℝ Real.sin x

@[simp]

theorem Real.deriv_sin : deriv Real.sin = Real.cos

theorem Real.hasStrictDerivAt_cos (x : ℝ) : HasStrictDerivAt Real.cos (-Real.sin x) x

theorem Real.hasDerivAt_cos (x : ℝ) : HasDerivAt Real.cos (-Real.sin x) x

theorem Real.contDiff_cos {n : ℕ∞} : ContDiff ℝ n Real.cos

theorem Real.differentiable_cos : Differentiable ℝ Real.cos

theorem Real.differentiableAt_cos {x : ℝ} : DifferentiableAt ℝ Real.cos x

theorem Real.deriv_cos {x : ℝ} : deriv Real.cos x = -Real.sin x

@[simp]

theorem Real.deriv_cos' : deriv Real.cos = fun x => -Real.sin x

theorem Real.hasStrictDerivAt_sinh (x : ℝ) : HasStrictDerivAt Real.sinh (Real.cosh x) x

theorem Real.hasDerivAt_sinh (x : ℝ) : HasDerivAt Real.sinh (Real.cosh x) x

theorem Real.contDiff_sinh {n : ℕ∞} : ContDiff ℝ n Real.sinh

theorem Real.differentiable_sinh : Differentiable ℝ Real.sinh

theorem Real.differentiableAt_sinh {x : ℝ} : DifferentiableAt ℝ Real.sinh x

@[simp]

theorem Real.deriv_sinh : deriv Real.sinh = Real.cosh

theorem Real.hasStrictDerivAt_cosh (x : ℝ) : HasStrictDerivAt Real.cosh (Real.sinh x) x

theorem Real.hasDerivAt_cosh (x : ℝ) : HasDerivAt Real.cosh (Real.sinh x) x

theorem Real.contDiff_cosh {n : ℕ∞} : ContDiff ℝ n Real.cosh

theorem Real.differentiable_cosh : Differentiable ℝ Real.cosh

theorem Real.differentiableAt_cosh {x : ℝ} : DifferentiableAt ℝ Real.cosh x

@[simp]

theorem Real.deriv_cosh : deriv Real.cosh = Real.sinh

theorem Real.sinh_strictMono : StrictMono Real.sinh

sinh is strictly monotone.

theorem Real.sinh_injective : Function.Injective Real.sinh

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

@[simp]

theorem Real.sinh_inj {x : ℝ} {y : ℝ} : Real.sinh x = Real.sinh y ↔ x = y

@[simp]

theorem Real.sinh_le_sinh {x : ℝ} {y : ℝ} : Real.sinh x ≤ Real.sinh y ↔ x ≤ y

@[simp]

theorem Real.sinh_lt_sinh {x : ℝ} {y : ℝ} : Real.sinh x < Real.sinh y ↔ x < y

@[simp]

theorem Real.sinh_pos_iff {x : ℝ} : 0 < Real.sinh x ↔ 0 < x

@[simp]

theorem Real.sinh_nonpos_iff {x : ℝ} : Real.sinh x ≤ 0 ↔ x ≤ 0

@[simp]

theorem Real.sinh_neg_iff {x : ℝ} : Real.sinh x < 0 ↔ x < 0

@[simp]

theorem Real.sinh_nonneg_iff {x : ℝ} : 0 ≤ Real.sinh x ↔ 0 ≤ x

theorem Real.abs_sinh (x : ℝ) : |Real.sinh x| = Real.sinh |x|

theorem Real.cosh_strictMonoOn : StrictMonoOn Real.cosh (Set.Ici 0)

@[simp]

theorem Real.cosh_le_cosh {x : ℝ} {y : ℝ} : Real.cosh x ≤ Real.cosh y ↔ |x| ≤ |y|

@[simp]

theorem Real.cosh_lt_cosh {x : ℝ} {y : ℝ} : Real.cosh x < Real.cosh y ↔ |x| < |y|

@[simp]

theorem Real.one_le_cosh (x : ℝ) : 1 ≤ Real.cosh x

@[simp]

theorem Real.one_lt_cosh {x : ℝ} : 1 < Real.cosh x ↔ x ≠ 0

theorem Real.sinh_sub_id_strictMono : StrictMono fun x => Real.sinh x - x

@[simp]

theorem Real.self_le_sinh_iff {x : ℝ} : x ≤ Real.sinh x ↔ 0 ≤ x

@[simp]

theorem Real.sinh_le_self_iff {x : ℝ} : Real.sinh x ≤ x ↔ x ≤ 0

@[simp]

theorem Real.self_lt_sinh_iff {x : ℝ} : x < Real.sinh x ↔ 0 < x

@[simp]

theorem Real.sinh_lt_self_iff {x : ℝ} : Real.sinh x < x ↔ x < 0

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

Real.cos

theorem HasStrictDerivAt.cos {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x

theorem HasDerivAt.cos {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x

theorem HasDerivWithinAt.cos {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') s x

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 => Real.cos (f x)) s x = -Real.sin (f x) * derivWithin f s x

@[simp]

theorem deriv_cos {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) : deriv (fun x => Real.cos (f x)) x = -Real.sin (f x) * deriv f x

Real.sin

theorem HasStrictDerivAt.sin {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') x

theorem HasDerivAt.sin {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') x

theorem HasDerivWithinAt.sin {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') s x

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 => Real.sin (f x)) s x = Real.cos (f x) * derivWithin f s x

@[simp]

theorem deriv_sin {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) : deriv (fun x => Real.sin (f x)) x = Real.cos (f x) * deriv f x

Real.cosh

theorem HasStrictDerivAt.cosh {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') x

theorem HasDerivAt.cosh {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') x

theorem HasDerivWithinAt.cosh {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') s x

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 => Real.cosh (f x)) s x = Real.sinh (f x) * derivWithin f s x

@[simp]

theorem deriv_cosh {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) : deriv (fun x => Real.cosh (f x)) x = Real.sinh (f x) * deriv f x

Real.sinh

theorem HasStrictDerivAt.sinh {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') x

theorem HasDerivAt.sinh {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} (hf : HasDerivAt f f' x) : HasDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') x

theorem HasDerivWithinAt.sinh {f : ℝ → ℝ} {f' : ℝ} {x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') s x

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 => Real.sinh (f x)) s x = Real.cosh (f x) * derivWithin f s x

@[simp]

theorem deriv_sinh {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x) : deriv (fun x => 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

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 => Real.cos (f x)) (-Real.sin (f x) • f') x

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 => Real.cos (f x)) (-Real.sin (f x) • f') x

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 => Real.cos (f x)) (-Real.sin (f x) • f') s x

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 => Real.cos (f x)) s x

@[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 => Real.cos (f x)) x

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 => Real.cos (f x)) s

@[simp]

theorem Differentiable.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.cos (f x)

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 => Real.cos (f x)) s x = -Real.sin (f x) • fderivWithin ℝ f s x

@[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 => Real.cos (f x)) x = -Real.sin (f x) • fderiv ℝ f x

theorem ContDiff.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.cos (f x)

theorem ContDiffAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.cos (f x)) x

theorem ContDiffOn.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.cos (f x)) s

theorem ContDiffWithinAt.cos {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) : ContDiffWithinAt ℝ n (fun x => Real.cos (f x)) s x

Real.sin

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 => Real.sin (f x)) (Real.cos (f x) • f') x

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 => Real.sin (f x)) (Real.cos (f x) • f') x

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 => Real.sin (f x)) (Real.cos (f x) • f') s x

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 => Real.sin (f x)) s x

@[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 => Real.sin (f x)) x

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 => Real.sin (f x)) s

@[simp]

theorem Differentiable.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.sin (f x)

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 => Real.sin (f x)) s x = Real.cos (f x) • fderivWithin ℝ f s x

@[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 => Real.sin (f x)) x = Real.cos (f x) • fderiv ℝ f x

theorem ContDiff.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.sin (f x)

theorem ContDiffAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.sin (f x)) x

theorem ContDiffOn.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.sin (f x)) s

theorem ContDiffWithinAt.sin {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) : ContDiffWithinAt ℝ n (fun x => Real.sin (f x)) s x

Real.cosh

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 => Real.cosh (f x)) (Real.sinh (f x) • f') x

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 => Real.cosh (f x)) (Real.sinh (f x) • f') x

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 => Real.cosh (f x)) (Real.sinh (f x) • f') s x

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 => Real.cosh (f x)) s x

@[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 => Real.cosh (f x)) x

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 => Real.cosh (f x)) s

@[simp]

theorem Differentiable.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.cosh (f x)

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 => Real.cosh (f x)) s x = Real.sinh (f x) • fderivWithin ℝ f s x

@[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 => Real.cosh (f x)) x = Real.sinh (f x) • fderiv ℝ f x

theorem ContDiff.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.cosh (f x)

theorem ContDiffAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.cosh (f x)) x

theorem ContDiffOn.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.cosh (f x)) s

theorem ContDiffWithinAt.cosh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) : ContDiffWithinAt ℝ n (fun x => Real.cosh (f x)) s x

Real.sinh

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 => Real.sinh (f x)) (Real.cosh (f x) • f') x

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 => Real.sinh (f x)) (Real.cosh (f x) • f') x

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 => Real.sinh (f x)) (Real.cosh (f x) • f') s x

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 => Real.sinh (f x)) s x

@[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 => Real.sinh (f x)) x

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 => Real.sinh (f x)) s

@[simp]

theorem Differentiable.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.sinh (f x)

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 => Real.sinh (f x)) s x = Real.cosh (f x) • fderivWithin ℝ f s x

@[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 => Real.sinh (f x)) x = Real.cosh (f x) • fderiv ℝ f x

theorem ContDiff.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.sinh (f x)

theorem ContDiffAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.sinh (f x)) x

theorem ContDiffOn.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.sinh (f x)) s

theorem ContDiffWithinAt.sinh {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) : ContDiffWithinAt ℝ n (fun x => Real.sinh (f x)) s x