theorem extracted_formal_statement_21.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let e := fun t ω => rexp (t * X ω);
  let e' := fun t ω => rexp (t * X ω) * X ω;
  Integrable (e' t) μ ∧ HasDerivAt (fun t => ∫ (x : Ω), e t x ∂μ) (∫ (x : Ω), e' t x ∂μ) t →
    Integrable (fun ω => rexp (t * X ω) * X ω) μ →
      HasDerivAt (fun t => ∫ (x : Ω), rexp (t * X x) ∂μ) (∫ (x : Ω), rexp (t * X x) * X x ∂μ) t := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
this : Integrable (e' t) μ ∧ HasDerivAt (fun t => ∫ (x : Ω), e t x ∂μ) (∫ (x : Ω), e' t x ∂μ) t
hint : Integrable (fun ω => rexp (t * X ω) * X ω) μ
⊢ HasDerivAt (fun t => ∫ (x : Ω), rexp (t * X x) ∂μ) (∫ (x : Ω), rexp (t * X x) * X x ∂μ) t

Tactic: apply this.2

State after:

Relevant lemmas: