theorem extracted_formal_statement_90.{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)
  (ω : Ω) (x : ℝ) (a_1 : x ∈ Metric.ball t (|t| + 1))
  (this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x)
  (h_original_goal : HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω ^ 2) x) :
  HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω * X ω) x := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
ω : Ω
x : ℝ
a✝ : x ∈ Metric.ball t (|t| + 1)
this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x
⊢ HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω ^ 2) x

Tactic: rw [(by ring : rexp (x * X ω) * X ω ^ 2 = (rexp (x * X ω) * X ω) * X ω)]

State after:
[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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ω : Ω
 x : ℝ
 a✝ : x ∈ Metric.ball t (|t| + 1)
 this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x
 ⊢ HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω * X ω) x]

Relevant lemmas: