ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- derivWithin_csin
theorem derivWithin_csin (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 
Arguments: {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x)
Goal: derivWithin (fun x => Complex.sin (f x)) s x = Complex.cos (f x) * derivWithin f s x
-- deriv_csin
@[simp]
theorem deriv_csin (hc : DifferentiableAt ℂ f x) :
    deriv (fun x => Complex.sin (f x)) x = Complex.cos (f x) * deriv f x 
Arguments: {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x)
Goal: deriv (fun x => Complex.sin (f x)) x = Complex.cos (f x) * deriv f x
-- HasStrictDerivAt.ccosh
theorem HasStrictDerivAt.ccosh (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') x
-- HasDerivAt.ccosh
theorem HasDerivAt.ccosh (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') x
-- HasDerivWithinAt.ccosh
theorem HasDerivWithinAt.ccosh (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') s x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) * f') s x
-- derivWithin_ccosh
theorem derivWithin_ccosh (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 
Arguments: {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x)
Goal: derivWithin (fun x => Complex.cosh (f x)) s x = Complex.sinh (f x) * derivWithin f s x
-- deriv_ccosh
@[simp]
theorem deriv_ccosh (hc : DifferentiableAt ℂ f x) :
    deriv (fun x => Complex.cosh (f x)) x = Complex.sinh (f x) * deriv f x 
Arguments: {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x)
Goal: deriv (fun x => Complex.cosh (f x)) x = Complex.sinh (f x) * deriv f x
-- HasStrictDerivAt.csinh
theorem HasStrictDerivAt.csinh (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') x
-- HasDerivAt.csinh
theorem HasDerivAt.csinh (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') x
-- HasDerivWithinAt.csinh
theorem HasDerivWithinAt.csinh (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') s x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) * f') s x
-- derivWithin_csinh
theorem derivWithin_csinh (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 
Arguments: {f : ℂ → ℂ} {x : ℂ} {s : Set ℂ} (hf : DifferentiableWithinAt ℂ f s x) (hxs : UniqueDiffWithinAt ℂ s x)
Goal: derivWithin (fun x => Complex.sinh (f x)) s x = Complex.cosh (f x) * derivWithin f s x
-- deriv_csinh
@[simp]
theorem deriv_csinh (hc : DifferentiableAt ℂ f x) :
    deriv (fun x => Complex.sinh (f x)) x = Complex.cosh (f x) * deriv f x 
Arguments: {f : ℂ → ℂ} {x : ℂ} (hc : DifferentiableAt ℂ f x)
Goal: deriv (fun x => Complex.sinh (f x)) x = Complex.cosh (f x) * deriv f x
-- HasStrictFDerivAt.ccos
theorem HasStrictFDerivAt.ccos (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) • f') x
-- HasFDerivAt.ccos
theorem HasFDerivAt.ccos (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) • f') x
-- HasFDerivWithinAt.ccos
theorem HasFDerivWithinAt.ccos (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) • f') s x 
Arguments: {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)
Goal: HasFDerivWithinAt (fun x => Complex.cos (f x)) (-Complex.sin (f x) • f') s x
-- DifferentiableWithinAt.ccos
theorem DifferentiableWithinAt.ccos (hf : DifferentiableWithinAt ℂ f s x) :
    DifferentiableWithinAt ℂ (fun x => Complex.cos (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x)
Goal: DifferentiableWithinAt ℂ (fun x => Complex.cos (f x)) s x
-- DifferentiableAt.ccos
@[simp]
theorem DifferentiableAt.ccos (hc : DifferentiableAt ℂ f x) :
    DifferentiableAt ℂ (fun x => Complex.cos (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: DifferentiableAt ℂ (fun x => Complex.cos (f x)) x
-- DifferentiableOn.ccos
theorem DifferentiableOn.ccos (hc : DifferentiableOn ℂ f s) :
    DifferentiableOn ℂ (fun x => Complex.cos (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s)
Goal: DifferentiableOn ℂ (fun x => Complex.cos (f x)) s
-- Differentiable.ccos
@[simp]
theorem Differentiable.ccos (hc : Differentiable ℂ f) :
    Differentiable ℂ fun x => Complex.cos (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f)
Goal: Differentiable ℂ fun x => Complex.cos (f x)
-- fderivWithin_ccos
theorem fderivWithin_ccos (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 
Arguments: {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)
Goal: fderivWithin ℂ (fun x => Complex.cos (f x)) s x = -Complex.sin (f x) • fderivWithin ℂ f s x
-- fderiv_ccos
@[simp]
theorem fderiv_ccos (hc : DifferentiableAt ℂ f x) :
    fderiv ℂ (fun x => Complex.cos (f x)) x = -Complex.sin (f x) • fderiv ℂ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: fderiv ℂ (fun x => Complex.cos (f x)) x = -Complex.sin (f x) • fderiv ℂ f x
-- ContDiff.ccos
theorem ContDiff.ccos {n} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.cos (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f)
Goal: ContDiff ℂ n fun x => Complex.cos (f x)
-- ContDiffAt.ccos
theorem ContDiffAt.ccos {n} (hf : ContDiffAt ℂ n f x) :
    ContDiffAt ℂ n (fun x => Complex.cos (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x)
Goal: ContDiffAt ℂ n (fun x => Complex.cos (f x)) x
-- ContDiffOn.ccos
theorem ContDiffOn.ccos {n} (hf : ContDiffOn ℂ n f s) :
    ContDiffOn ℂ n (fun x => Complex.cos (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s)
Goal: ContDiffOn ℂ n (fun x => Complex.cos (f x)) s
-- ContDiffWithinAt.ccos
theorem ContDiffWithinAt.ccos {n} (hf : ContDiffWithinAt ℂ n f s x) :
    ContDiffWithinAt ℂ n (fun x => Complex.cos (f x)) s x 
Arguments: {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)
Goal: ContDiffWithinAt ℂ n (fun x => Complex.cos (f x)) s x
-- HasStrictFDerivAt.csin
theorem HasStrictFDerivAt.csin (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Complex.sin (f x)) (Complex.cos (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Complex.sin (f x)) (Complex.cos (f x) • f') x
-- HasFDerivAt.csin
theorem HasFDerivAt.csin (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Complex.sin (f x)) (Complex.cos (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Complex.sin (f x)) (Complex.cos (f x) • f') x
-- HasFDerivWithinAt.csin
theorem HasFDerivWithinAt.csin (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Complex.sin (f x)) (Complex.cos (f x) • f') s x 
Arguments: {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)
Goal: HasFDerivWithinAt (fun x => Complex.sin (f x)) (Complex.cos (f x) • f') s x
-- DifferentiableWithinAt.csin
theorem DifferentiableWithinAt.csin (hf : DifferentiableWithinAt ℂ f s x) :
    DifferentiableWithinAt ℂ (fun x => Complex.sin (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x)
Goal: DifferentiableWithinAt ℂ (fun x => Complex.sin (f x)) s x
-- DifferentiableAt.csin
@[simp]
theorem DifferentiableAt.csin (hc : DifferentiableAt ℂ f x) :
    DifferentiableAt ℂ (fun x => Complex.sin (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: DifferentiableAt ℂ (fun x => Complex.sin (f x)) x
-- DifferentiableOn.csin
theorem DifferentiableOn.csin (hc : DifferentiableOn ℂ f s) :
    DifferentiableOn ℂ (fun x => Complex.sin (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s)
Goal: DifferentiableOn ℂ (fun x => Complex.sin (f x)) s
-- Differentiable.csin
@[simp]
theorem Differentiable.csin (hc : Differentiable ℂ f) :
    Differentiable ℂ fun x => Complex.sin (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f)
Goal: Differentiable ℂ fun x => Complex.sin (f x)
-- fderivWithin_csin
theorem fderivWithin_csin (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 
Arguments: {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)
Goal: fderivWithin ℂ (fun x => Complex.sin (f x)) s x = Complex.cos (f x) • fderivWithin ℂ f s x
-- fderiv_csin
@[simp]
theorem fderiv_csin (hc : DifferentiableAt ℂ f x) :
    fderiv ℂ (fun x => Complex.sin (f x)) x = Complex.cos (f x) • fderiv ℂ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: fderiv ℂ (fun x => Complex.sin (f x)) x = Complex.cos (f x) • fderiv ℂ f x
-- ContDiff.csin
theorem ContDiff.csin {n} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.sin (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f)
Goal: ContDiff ℂ n fun x => Complex.sin (f x)
-- ContDiffAt.csin
theorem ContDiffAt.csin {n} (hf : ContDiffAt ℂ n f x) :
    ContDiffAt ℂ n (fun x => Complex.sin (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x)
Goal: ContDiffAt ℂ n (fun x => Complex.sin (f x)) x
-- ContDiffOn.csin
theorem ContDiffOn.csin {n} (hf : ContDiffOn ℂ n f s) :
    ContDiffOn ℂ n (fun x => Complex.sin (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s)
Goal: ContDiffOn ℂ n (fun x => Complex.sin (f x)) s
-- ContDiffWithinAt.csin
theorem ContDiffWithinAt.csin {n} (hf : ContDiffWithinAt ℂ n f s x) :
    ContDiffWithinAt ℂ n (fun x => Complex.sin (f x)) s x 
Arguments: {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)
Goal: ContDiffWithinAt ℂ n (fun x => Complex.sin (f x)) s x
-- HasStrictFDerivAt.ccosh
theorem HasStrictFDerivAt.ccosh (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) • f') x
-- HasFDerivAt.ccosh
theorem HasFDerivAt.ccosh (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) • f') x
-- HasFDerivWithinAt.ccosh
theorem HasFDerivWithinAt.ccosh (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) • f') s x 
Arguments: {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)
Goal: HasFDerivWithinAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) • f') s x
-- DifferentiableWithinAt.ccosh
theorem DifferentiableWithinAt.ccosh (hf : DifferentiableWithinAt ℂ f s x) :
    DifferentiableWithinAt ℂ (fun x => Complex.cosh (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x)
Goal: DifferentiableWithinAt ℂ (fun x => Complex.cosh (f x)) s x
-- DifferentiableAt.ccosh
@[simp]
theorem DifferentiableAt.ccosh (hc : DifferentiableAt ℂ f x) :
    DifferentiableAt ℂ (fun x => Complex.cosh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: DifferentiableAt ℂ (fun x => Complex.cosh (f x)) x
-- DifferentiableOn.ccosh
theorem DifferentiableOn.ccosh (hc : DifferentiableOn ℂ f s) :
    DifferentiableOn ℂ (fun x => Complex.cosh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s)
Goal: DifferentiableOn ℂ (fun x => Complex.cosh (f x)) s
-- Differentiable.ccosh
@[simp]
theorem Differentiable.ccosh (hc : Differentiable ℂ f) :
    Differentiable ℂ fun x => Complex.cosh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f)
Goal: Differentiable ℂ fun x => Complex.cosh (f x)
-- fderivWithin_ccosh
theorem fderivWithin_ccosh (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 
Arguments: {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)
Goal: fderivWithin ℂ (fun x => Complex.cosh (f x)) s x = Complex.sinh (f x) • fderivWithin ℂ f s x
-- fderiv_ccosh
@[simp]
theorem fderiv_ccosh (hc : DifferentiableAt ℂ f x) :
    fderiv ℂ (fun x => Complex.cosh (f x)) x = Complex.sinh (f x) • fderiv ℂ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: fderiv ℂ (fun x => Complex.cosh (f x)) x = Complex.sinh (f x) • fderiv ℂ f x
-- ContDiff.ccosh
theorem ContDiff.ccosh {n} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.cosh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f)
Goal: ContDiff ℂ n fun x => Complex.cosh (f x)
-- ContDiffAt.ccosh
theorem ContDiffAt.ccosh {n} (hf : ContDiffAt ℂ n f x) :
    ContDiffAt ℂ n (fun x => Complex.cosh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x)
Goal: ContDiffAt ℂ n (fun x => Complex.cosh (f x)) x
-- ContDiffOn.ccosh
theorem ContDiffOn.ccosh {n} (hf : ContDiffOn ℂ n f s) :
    ContDiffOn ℂ n (fun x => Complex.cosh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s)
Goal: ContDiffOn ℂ n (fun x => Complex.cosh (f x)) s
-- ContDiffWithinAt.ccosh
theorem ContDiffWithinAt.ccosh {n} (hf : ContDiffWithinAt ℂ n f s x) :
    ContDiffWithinAt ℂ n (fun x => Complex.cosh (f x)) s x 
Arguments: {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)
Goal: ContDiffWithinAt ℂ n (fun x => Complex.cosh (f x)) s x
-- HasStrictFDerivAt.csinh
theorem HasStrictFDerivAt.csinh (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) • f') x
-- HasFDerivAt.csinh
theorem HasFDerivAt.csinh (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) • f') x
-- HasFDerivWithinAt.csinh
theorem HasFDerivWithinAt.csinh (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) • f') s x 
Arguments: {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)
Goal: HasFDerivWithinAt (fun x => Complex.sinh (f x)) (Complex.cosh (f x) • f') s x
-- DifferentiableWithinAt.csinh
theorem DifferentiableWithinAt.csinh (hf : DifferentiableWithinAt ℂ f s x) :
    DifferentiableWithinAt ℂ (fun x => Complex.sinh (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℂ f s x)
Goal: DifferentiableWithinAt ℂ (fun x => Complex.sinh (f x)) s x
-- DifferentiableAt.csinh
@[simp]
theorem DifferentiableAt.csinh (hc : DifferentiableAt ℂ f x) :
    DifferentiableAt ℂ (fun x => Complex.sinh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: DifferentiableAt ℂ (fun x => Complex.sinh (f x)) x
-- DifferentiableOn.csinh
theorem DifferentiableOn.csinh (hc : DifferentiableOn ℂ f s) :
    DifferentiableOn ℂ (fun x => Complex.sinh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (hc : DifferentiableOn ℂ f s)
Goal: DifferentiableOn ℂ (fun x => Complex.sinh (f x)) s
-- Differentiable.csinh
@[simp]
theorem Differentiable.csinh (hc : Differentiable ℂ f) :
    Differentiable ℂ fun x => Complex.sinh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (hc : Differentiable ℂ f)
Goal: Differentiable ℂ fun x => Complex.sinh (f x)
-- fderivWithin_csinh
theorem fderivWithin_csinh (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 
Arguments: {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)
Goal: fderivWithin ℂ (fun x => Complex.sinh (f x)) s x = Complex.cosh (f x) • fderivWithin ℂ f s x
-- fderiv_csinh
@[simp]
theorem fderiv_csinh (hc : DifferentiableAt ℂ f x) :
    fderiv ℂ (fun x => Complex.sinh (f x)) x = Complex.cosh (f x) • fderiv ℂ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (hc : DifferentiableAt ℂ f x)
Goal: fderiv ℂ (fun x => Complex.sinh (f x)) x = Complex.cosh (f x) • fderiv ℂ f x
-- ContDiff.csinh
theorem ContDiff.csinh {n} (h : ContDiff ℂ n f) : ContDiff ℂ n fun x => Complex.sinh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {n : ℕ∞} (h : ContDiff ℂ n f)
Goal: ContDiff ℂ n fun x => Complex.sinh (f x)
-- ContDiffAt.csinh
theorem ContDiffAt.csinh {n} (hf : ContDiffAt ℂ n f x) :
    ContDiffAt ℂ n (fun x => Complex.sinh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℂ n f x)
Goal: ContDiffAt ℂ n (fun x => Complex.sinh (f x)) x
-- ContDiffOn.csinh
theorem ContDiffOn.csinh {n} (hf : ContDiffOn ℂ n f s) :
    ContDiffOn ℂ n (fun x => Complex.sinh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℂ n f s)
Goal: ContDiffOn ℂ n (fun x => Complex.sinh (f x)) s
-- ContDiffWithinAt.csinh
theorem ContDiffWithinAt.csinh {n} (hf : ContDiffWithinAt ℂ n f s x) :
    ContDiffWithinAt ℂ n (fun x => Complex.sinh (f x)) s x 
Arguments: {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)
Goal: ContDiffWithinAt ℂ n (fun x => Complex.sinh (f x)) s x
-- Real.hasStrictDerivAt_sin
theorem hasStrictDerivAt_sin (x : ℝ) : HasStrictDerivAt sin (cos x) x 
Arguments: (x : ℝ)
Goal: HasStrictDerivAt Real.sin (Real.cos x) x
-- Real.hasDerivAt_sin
theorem hasDerivAt_sin (x : ℝ) : HasDerivAt sin (cos x) x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.sin (Real.cos x) x
-- Real.contDiff_sin
theorem contDiff_sin {n} : ContDiff ℝ n sin 
Arguments: {n : ℕ∞}
Goal: ContDiff ℝ n Real.sin
-- Real.differentiable_sin
theorem differentiable_sin : Differentiable ℝ sin 
Goal: Differentiable ℝ Real.sin
-- Real.differentiableAt_sin
theorem differentiableAt_sin : DifferentiableAt ℝ sin x 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.sin x
-- Real.deriv_sin
@[simp]
theorem deriv_sin : deriv sin = cos 
Goal: deriv Real.sin = Real.cos
-- Real.hasStrictDerivAt_cos
theorem hasStrictDerivAt_cos (x : ℝ) : HasStrictDerivAt cos (-sin x) x 
Arguments: (x : ℝ)
Goal: HasStrictDerivAt Real.cos (-Real.sin x) x
-- Real.hasDerivAt_cos
theorem hasDerivAt_cos (x : ℝ) : HasDerivAt cos (-sin x) x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.cos (-Real.sin x) x
-- Real.contDiff_cos
theorem contDiff_cos {n} : ContDiff ℝ n cos 
Arguments: {n : ℕ∞}
Goal: ContDiff ℝ n Real.cos
-- Real.differentiable_cos
theorem differentiable_cos : Differentiable ℝ cos 
Goal: Differentiable ℝ Real.cos
-- Real.differentiableAt_cos
theorem differentiableAt_cos : DifferentiableAt ℝ cos x 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.cos x
-- Real.deriv_cos
theorem deriv_cos : deriv cos x = -sin x 
Arguments: {x : ℝ}
Goal: deriv Real.cos x = -Real.sin x
-- Real.deriv_cos'
@[simp]
theorem deriv_cos' : deriv cos = fun x => -sin x 
Goal: deriv Real.cos = fun x => -Real.sin x
-- Real.hasStrictDerivAt_sinh
theorem hasStrictDerivAt_sinh (x : ℝ) : HasStrictDerivAt sinh (cosh x) x 
Arguments: (x : ℝ)
Goal: HasStrictDerivAt Real.sinh (Real.cosh x) x
-- Real.hasDerivAt_sinh
theorem hasDerivAt_sinh (x : ℝ) : HasDerivAt sinh (cosh x) x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.sinh (Real.cosh x) x
-- Real.contDiff_sinh
theorem contDiff_sinh {n} : ContDiff ℝ n sinh 
Arguments: {n : ℕ∞}
Goal: ContDiff ℝ n Real.sinh
-- Real.differentiable_sinh
theorem differentiable_sinh : Differentiable ℝ sinh 
Goal: Differentiable ℝ Real.sinh