theorem extracted_formal_statement_91.{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) :
  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 * X ω) x

Tactic: exact HasDerivAt.mul_const this (X ω)

State after:

Relevant lemmas:
HasDerivAt.mul_const : ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [inst_1 : NormedRing 𝔸]
  [inst_2 : NormedAlgebra 𝕜 𝔸] {c : 𝕜 → 𝔸} {c' : 𝔸},
  HasDerivAt c c' x → ∀ (d : 𝔸), HasDerivAt (fun y => c y * d) (c' * d) x