theorem extracted_formal_statement_15.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] [inst_1 : NeZero μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ)
  (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let f := fun t => cgf X μ t;
  ∀ (t : ℝ),
    (∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) =
        (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ →
      HasDerivAt f (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) t := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
f : ℝ → ℝ := fun t => cgf X μ t
f' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω
t : ℝ
g : ℝ → ℝ := fun t => ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
g' : ℝ → ℝ := fun t => ∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ
r0 :
  (∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
⊢ HasDerivAt f (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) t

Tactic: simp only [id_eq, f'] at r0
    rw [r0]
    apply
  HasDerivAt.log (tilt_first_deriv _ _ _ hX h (integrable_deriv_expt t a b hX h))
    (mgf_pos' (NeZero.ne μ) (integrable_expt_bound hX h)).ne'

State after:

Relevant lemmas:
error