theorem extracted_formal_statement_48.{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;
  ∀ (ω : Ω), ∀ x ∈ Metric.ball t (|t| + 1), HasDerivAt (fun x => e x ω) (e' x ω) x := sorry

State before:
case h_diff
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
ω : Ω
x : ℝ
a✝ : x ∈ Metric.ball t (|t| + 1)
⊢ HasDerivAt (fun x => e x ω) (e' x ω) x

Tactic: exact HasDerivAt.exp (hasDerivAt_mul_const (X ω))

State after:

Relevant lemmas:
hasDerivAt_mul_const : ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜} (c : 𝕜),
  HasDerivAt (fun x => x * c) c x
HasDerivAt.exp : ∀ {f : ℝ → ℝ} {f' x : ℝ}, HasDerivAt f f' x → HasDerivAt (fun x => rexp (f x)) (rexp (f x) * f') x