File: Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean

full_name: Complex.hasDerivAt_exp
code: /-- The complex exponential is everywhere differentiable, with the derivative `exp x`. -/
theorem hasDerivAt_exp (x : ℂ) : HasDerivAt exp (exp x) x 
goal: HasDerivAt Complex.exp (Complex.exp x) x

full_name: Complex.differentiable_exp
code: theorem differentiable_exp : Differentiable 𝕜 exp 
goal: Differentiable 𝕜 Complex.exp

full_name: Complex.differentiableAt_exp
code: theorem differentiableAt_exp {x : ℂ} : DifferentiableAt 𝕜 exp x 
goal: DifferentiableAt 𝕜 Complex.exp x

full_name: Complex.deriv_exp
code: @[simp]
theorem deriv_exp : deriv exp = exp 
goal: deriv Complex.exp = Complex.exp

full_name: Complex.iter_deriv_exp
code: @[simp]
theorem iter_deriv_exp : ∀ n : ℕ, deriv^[n] exp = exp
  
goal: deriv^[n] Complex.exp = Complex.exp

full_name: Complex.contDiff_exp
code: theorem contDiff_exp : ∀ {n}, ContDiff 𝕜 n exp 
goal: ContDiff 𝕜 n Complex.exp

full_name: Complex.hasStrictDerivAt_exp
code: theorem hasStrictDerivAt_exp (x : ℂ) : HasStrictDerivAt exp (exp x) x 
goal: HasStrictDerivAt Complex.exp (Complex.exp x) x

full_name: Complex.hasStrictFDerivAt_exp_real
code: theorem hasStrictFDerivAt_exp_real (x : ℂ) : HasStrictFDerivAt exp (exp x • (1 : ℂ →L[ℝ] ℂ)) x 
goal: HasStrictFDerivAt Complex.exp (Complex.exp x • 1) x

full_name: HasStrictDerivAt.cexp
code: theorem HasStrictDerivAt.cexp (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x 
goal: HasStrictDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x

full_name: HasDerivAt.cexp
code: theorem HasDerivAt.cexp (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x 
goal: HasDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') x

full_name: HasDerivWithinAt.cexp
code: theorem HasDerivWithinAt.cexp (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') s x 
goal: HasDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) * f') s x

full_name: derivWithin_cexp
code: theorem derivWithin_cexp (hf : DifferentiableWithinAt 𝕜 f s x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => Complex.exp (f x)) s x = Complex.exp (f x) * derivWithin f s x 
goal: derivWithin (fun x => Complex.exp (f x)) s x = Complex.exp (f x) * derivWithin f s x

full_name: deriv_cexp
code: @[simp]
theorem deriv_cexp (hc : DifferentiableAt 𝕜 f x) :
    deriv (fun x => Complex.exp (f x)) x = Complex.exp (f x) * deriv f x 
goal: deriv (fun x => Complex.exp (f x)) x = Complex.exp (f x) * deriv f x

full_name: HasStrictFDerivAt.cexp
code: theorem HasStrictFDerivAt.cexp (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x 
goal: HasStrictFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x

full_name: HasFDerivWithinAt.cexp
code: theorem HasFDerivWithinAt.cexp (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') s x 
goal: HasFDerivWithinAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') s x

full_name: HasFDerivAt.cexp
code: theorem HasFDerivAt.cexp (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x 
goal: HasFDerivAt (fun x => Complex.exp (f x)) (Complex.exp (f x) • f') x

full_name: DifferentiableWithinAt.cexp
code: theorem DifferentiableWithinAt.cexp (hf : DifferentiableWithinAt 𝕜 f s x) :
    DifferentiableWithinAt 𝕜 (fun x => Complex.exp (f x)) s x 
goal: DifferentiableWithinAt 𝕜 (fun x => Complex.exp (f x)) s x

full_name: DifferentiableAt.cexp
code: @[simp]
theorem DifferentiableAt.cexp (hc : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜 (fun x => Complex.exp (f x)) x 
goal: DifferentiableAt 𝕜 (fun x => Complex.exp (f x)) x

full_name: DifferentiableOn.cexp
code: theorem DifferentiableOn.cexp (hc : DifferentiableOn 𝕜 f s) :
    DifferentiableOn 𝕜 (fun x => Complex.exp (f x)) s 
goal: DifferentiableOn 𝕜 (fun x => Complex.exp (f x)) s

full_name: Differentiable.cexp
code: @[simp]
theorem Differentiable.cexp (hc : Differentiable 𝕜 f) :
    Differentiable 𝕜 fun x => Complex.exp (f x) 
goal: Differentiable 𝕜 fun x => Complex.exp (f x)

full_name: ContDiff.cexp
code: theorem ContDiff.cexp {n} (h : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => Complex.exp (f x) 
goal: ContDiff 𝕜 n fun x => Complex.exp (f x)

full_name: ContDiffAt.cexp
code: theorem ContDiffAt.cexp {n} (hf : ContDiffAt 𝕜 n f x) :
    ContDiffAt 𝕜 n (fun x => Complex.exp (f x)) x 
goal: ContDiffAt 𝕜 n (fun x => Complex.exp (f x)) x

full_name: ContDiffOn.cexp
code: theorem ContDiffOn.cexp {n} (hf : ContDiffOn 𝕜 n f s) :
    ContDiffOn 𝕜 n (fun x => Complex.exp (f x)) s 
goal: ContDiffOn 𝕜 n (fun x => Complex.exp (f x)) s

full_name: ContDiffWithinAt.cexp
code: theorem ContDiffWithinAt.cexp {n} (hf : ContDiffWithinAt 𝕜 n f s x) :
    ContDiffWithinAt 𝕜 n (fun x => Complex.exp (f x)) s x 
goal: ContDiffWithinAt 𝕜 n (fun x => Complex.exp (f x)) s x

full_name: iteratedDeriv_cexp_const_mul
code: @[simp]
theorem iteratedDeriv_cexp_const_mul (n : ℕ) (c : ℂ) :
    (iteratedDeriv n fun s : ℂ => exp (c * s)) = fun s => c ^ n * exp (c * s) 
goal: (iteratedDeriv n fun s => Complex.exp (c * s)) = fun s => c ^ n * Complex.exp (c * s)

full_name: Real.hasStrictDerivAt_exp
code: theorem hasStrictDerivAt_exp (x : ℝ) : HasStrictDerivAt exp (exp x) x 
goal: HasStrictDerivAt Real.exp (Real.exp x) x

full_name: Real.hasDerivAt_exp
code: theorem hasDerivAt_exp (x : ℝ) : HasDerivAt exp (exp x) x 
goal: HasDerivAt Real.exp (Real.exp x) x

full_name: Real.contDiff_exp
code: theorem contDiff_exp {n} : ContDiff ℝ n exp 
goal: ContDiff ℝ n Real.exp

full_name: Real.differentiable_exp
code: theorem differentiable_exp : Differentiable ℝ exp 
goal: Differentiable ℝ Real.exp

full_name: Real.differentiableAt_exp
code: theorem differentiableAt_exp : DifferentiableAt ℝ exp x 
goal: DifferentiableAt ℝ Real.exp x

full_name: Real.deriv_exp
code: @[simp]
theorem deriv_exp : deriv exp = exp 
goal: deriv Real.exp = Real.exp

full_name: Real.iter_deriv_exp
code: @[simp]
theorem iter_deriv_exp : ∀ n : ℕ, deriv^[n] exp = exp
  
goal: deriv^[n] Real.exp = Real.exp

full_name: HasStrictDerivAt.exp
code: theorem HasStrictDerivAt.exp (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x 
goal: HasStrictDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x

full_name: HasDerivAt.exp
code: theorem HasDerivAt.exp (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x 
goal: HasDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x

full_name: HasDerivWithinAt.exp
code: theorem HasDerivWithinAt.exp (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') s x 
goal: HasDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') s x

full_name: derivWithin_exp
code: theorem derivWithin_exp (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => Real.exp (f x)) s x = Real.exp (f x) * derivWithin f s x 
goal: derivWithin (fun x => Real.exp (f x)) s x = Real.exp (f x) * derivWithin f s x

full_name: deriv_exp
code: @[simp]
theorem deriv_exp (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => Real.exp (f x)) x = Real.exp (f x) * deriv f x 
goal: deriv (fun x => Real.exp (f x)) x = Real.exp (f x) * deriv f x

full_name: ContDiff.exp
code: theorem ContDiff.exp {n} (hf : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.exp (f x) 
goal: ContDiff ℝ n fun x => Real.exp (f x)

full_name: ContDiffAt.exp
code: theorem ContDiffAt.exp {n} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.exp (f x)) x 
goal: ContDiffAt ℝ n (fun x => Real.exp (f x)) x

full_name: ContDiffOn.exp
code: theorem ContDiffOn.exp {n} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.exp (f x)) s 
goal: ContDiffOn ℝ n (fun x => Real.exp (f x)) s

full_name: ContDiffWithinAt.exp
code: theorem ContDiffWithinAt.exp {n} (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => Real.exp (f x)) s x 
goal: ContDiffWithinAt ℝ n (fun x => Real.exp (f x)) s x

full_name: HasFDerivWithinAt.exp
code: theorem HasFDerivWithinAt.exp (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') s x 
goal: HasFDerivWithinAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') s x

full_name: HasFDerivAt.exp
code: theorem HasFDerivAt.exp (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x 
goal: HasFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x

full_name: HasStrictFDerivAt.exp
code: theorem HasStrictFDerivAt.exp (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x 
goal: HasStrictFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x

full_name: DifferentiableWithinAt.exp
code: theorem DifferentiableWithinAt.exp (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.exp (f x)) s x 
goal: DifferentiableWithinAt ℝ (fun x => Real.exp (f x)) s x

full_name: DifferentiableAt.exp
code: @[simp]
theorem DifferentiableAt.exp (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => Real.exp (f x)) x 
goal: DifferentiableAt ℝ (fun x => Real.exp (f x)) x

full_name: DifferentiableOn.exp
code: theorem DifferentiableOn.exp (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => Real.exp (f x)) s 
goal: DifferentiableOn ℝ (fun x => Real.exp (f x)) s

full_name: Differentiable.exp
code: @[simp]
theorem Differentiable.exp (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.exp (f x) 
goal: Differentiable ℝ fun x => Real.exp (f x)

full_name: fderivWithin_exp
code: theorem fderivWithin_exp (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => Real.exp (f x)) s x = Real.exp (f x) • fderivWithin ℝ f s x 
goal: fderivWithin ℝ (fun x => Real.exp (f x)) s x = Real.exp (f x) • fderivWithin ℝ f s x

full_name: fderiv_exp
code: @[simp]
theorem fderiv_exp (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => Real.exp (f x)) x = Real.exp (f x) • fderiv ℝ f x 
goal: fderiv ℝ (fun x => Real.exp (f x)) x = Real.exp (f x) • fderiv ℝ f x

full_name: iteratedDeriv_exp_const_mul
code: @[simp]
theorem iteratedDeriv_exp_const_mul (n : ℕ) (c : ℝ) :
    (iteratedDeriv n fun s => exp (c * s)) = fun s => c ^ n * exp (c * s) 
goal: (iteratedDeriv n fun s => Real.exp (c * s)) = fun s => c ^ n * Real.exp (c * s)